JOURNAL OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 11, Number 4, October 1998, Pages 899–965
S 0894-0347(98)00277-X
REGULARITY OF THE FREE BOUNDARY
FOR THE POROUS MEDIUM EQUATION
P. DASKALOPOULOS AND R. HAMILTON
Introduction
We consider the Cauchy problem for the porous medium equation
(
in Rn × [0, ∞),
ut = ∆um
u(x, 0) = u0
on Rn
in the range of exponents m > 1, with initial data u0 nonnegative, integrable and
compactly supported.
It is well known that this equation describes the evolution in time of various
diffusion processes, in particular the flow of a gas through a porous medium; u
represents the density, while f = m um−1 represents the pressure of the gas. The
function f satisfies the equation
ft = f ∆f + r(m) |Df |2
with r(m) = 1/(m − 1). When u = 0, then f = 0 and both of the above equations
become degenerate. This degeneracy results in the interesting phenomenon of the
finite speed of propagation: If the initial data u0 is compactly supported in Rn , the
solution u(·, t) will remain compactly supported for all time t.
In this work we will show that, under rather general assumptions on the initial
data, the free boundary
Γ = ∂ supp u
is a smooth surface when 0 < t < T , for some T > 0.
It is well known that if the initial data u0 is nonnegative, integrable and compactly supported, then the Cauchy problem for the porous medium equation admits
a unique solution on Rn ×(0, ∞) which has constant mass. However, since the equation becomes degenerate when u = 0, the solution is not expected to be smooth;
the optimal regularity for the density u has been shown to be Hölder continuous
and solutions are understood in the distributional sense. On the other hand, the
physical interpretation of the equation indicates that, under ideal conditions, the
free boundary should be a smooth surface and the pressure f a smooth function up
to the interface. However, this is not always the case: If support of the initial data
u0 is topologically complex, then advancing free-boundaries may hit each other
after a short time, creating singularities.
Received by the editors January 19, 1998.
1991 Mathematics Subject Classification. Primary 35Jxx.
Key words and phrases. Porous medium equation, free-boundary, C ∞ -regularity.
c
1998
American Mathematical Society
899
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
900
P. DASKALOPOULOS AND R. HAMILTON
Caffarelli and Friedman ([CF1], [CF2]) showed that the interface can always be
described by a Hölder continuous function t = S(x), x ∈
/ supp u0 , for any initial
data. In the one-dimensional case much more is known: It has been shown in [K]
and [CF1] that if the support of the initial data is an interval, the free boundary
consists of two Lipshitz continuous curves x = ζi (t), with ζ1 decreasing and ζ2
increasing. Moreover, there exist waiting times t∗i so that ζi (t) are constant for
t ≤ t∗i and each ζi is C 1 for t > t∗i . However at t = t∗i , ζi (t) may have a corner, so
Lipshitz is the optimal regularity (see [ACV]). Aronson and Vázquez ([AV]) and
independently Höllig and Kreiss ([HK]) showed that for t > t∗i the curves ζi are
smooth. Angenent ([A]) showed that for t > t∗i the curves ζi are real analytic.
In dimensions n ≥ 2 the regularity of the free boundary poses a much harder
question. Caffarelli, Vázquez and Wolanski ([CVW]) showed that, under the assumption that at time t = 0 the pressure f 0 ∈ C 1 (suppf 0 ) and Df 0 6= 0 along
∂ supp f 0 , a condition which ensures that the free boundary will start to move at
each point at t = 0, the interface can be described by a Lipshitz continuous function t = S(x), x ∈
/ suppf 0 . Caffarelli and Wolanski ([CW]) improved this result,
showing that, under the same hypotheses, the free boundary is a C 1,α surface.
However, it is not shown in [CW] that the pressure f is a C 1,α function up to the
free boundary.
In this work, assuming that the initial pressure f 0 is strictly positive in the
interior of a compact domain Ω in R2 , with f 0 = 0 on ∂Ω ∪ (Rn \ Ω), and denoting
by d the distance to the boundary of Ω, we obtain the following result:
Theorem (C ∞ -regularity of the boundary). If the functions f 0 , Df 0 and d D2 f 0 ,
restricted to the compact domain Ω, extend continuously up to the boundary Ω,
with extensions which are Hölder continuous on Ω of class C α (Ω), for some α > 0,
and Df 0 6= 0 along ∂Ω, then there exists a number T > 0 for which the initial value
problem
(
ft = f ∆f + r |Df |2 ,
(x, t) ∈ Rn × [0, T ],
0
f (x, 0) = f ,
x ∈ Rn ,
admits a solution f which is smooth up to the interface Γ, when 0 < t < T . In
particular, the free boundary Γ is a smooth surface, when 0 < t < T .
The paper is divided into three parts: In Part 1 we study the model linear
degenerate equation
ft = x (fxx + fyy ) + ν fx + g
with ν > 0 on the half-space x ≥ 0, and no extra conditions on f along the
boundary x = 0. The diffusion in this equation is governed by the Riemannian
metric ds2 = (dx2 + dy 2 )/2x. It is a simple observation that the equation admits
solutions which behave like x1−ν near the boundary x = 0. However, once these
solutions are ruled out, all other solutions are smooth. The basic idea in our
approach is to establish Schauder type coercive estimates for the solutions f of the
model equation, which are scaled according to the new metric ds. In Part 2 we
extend the results of Part 1 to a certain class of quasilinear degenerate evolution
equations. The proof of the regularity of the free boundary is given in Part 3. Using
a global change of coordinates, we transform the free boundary problem to a fixed
boundary problem for a degenerate quasilinear equation, which can be solved, in
appropriately defined Hölder spaces, using the results from Parts 1 and 2.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
901
Part I. The model linear degenerate equation
I.1. The results. The first part of the paper will be devoted to the study of the
model degenerate equation
ft = x (fxx + fyy ) + ν fx + g
with ν > 0 on the half-space x ≥ 0, and no extra conditions on f along the boundary
x = 0. The diffusion is governed by the Riemannian metric ds where
dx2 + dy 2
.
2x
We call this the cycloidal metric because its geodesics are cycloid curves. Its Laplace
operator is
ds2 =
∆s f = 2x (fxx + fyy ) .
Our evolution equation can be written with respect to the new metric as
1
ft = ∆s f + ν fx
2
which is diffusion in the cycloidal metric with transport velocity ν. The distance
between two points ( xy11 ) and ( xy22 ) in this metric is a function
x2
x1
,
s
y1
y2
which is equivalent to the function
|x1 − x2 | + |y1 − y2 |
x2
x1
p
,
= √
s
√
y1
y2
x1 + x2 + |y1 − y2 |
in the sense that
s≤Cs
and
s≤Cs
for some constant C. For the parabolic problem we use the parabolic distance
   
x1
x2
p
x2
x1






y1 , y2
s
,
+ |t1 − t2 | .
=s
y1
y2
t1
t2
In terms of this distance we can define Hölder continuity. We say a continuous
function g on a compact subset A of the half-space { (x, y, t) : x ≥ 0 } is Hölder
continuous with respect to the metric s if for all points P1 and P2 in A we have
|g(P1 ) − g(P2 )| ≤ C s[P1 , P2 ]α
and we define the Hölder semi-norm
kgkHsα (A) = sup |g(P1 ) − g(P2 )|/s[P1 , P2 ]α .
P1 6=P2
We also define the norm
kgkCsα (A) = kgkC 0 (A) + kgkHsα (A)
where as usual
kgkC 0 (A) = sup |g(P )| .
P
With this norm the space Csα (A) of Hölder continuous functions on A with respect
to the metric s is a Banach space (as usual).
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
902
P. DASKALOPOULOS AND R. HAMILTON
Now suppose the set A is the closure of its interior, and the function f on A has
continuous derivatives
ft , fx , fy , fxx , fxy , fyy
in the interior of A, and that
ft , fx , fy
and
xfxx , xfxy , xfyy
extend continuously to the boundary, and the extensions are Hölder continuous on
A of class Csα (A) as before. Let Cs2+α (A) be the Banach space of all such functions
with norm
kf kCs2+α (A) = kf kCsα(A) + kfx kCsα (A) + kfy kCsα (A) + kft kCsα (A)
+ kxfxx kCsα (A) + kxfxy kCsα (A) + kxfyy kCsα (A) .
The operator L0 defined by
L0 = ft − x(fxx + fyy ) − νfx
defines a continuous linear map
L0 : Cs2+α (A) → Csα (A) .
We can extend these definitions to spaces of higher order derivatives. Let k be a
positive integer and let A be a subset of the half-space x ≥ 0 as above. We denote
by Csk,α (A) the space of all functions g whose k-th order derivatives Dxi Dyj Dtl g,
i + j + l = k, exist and belong to the space Csα (A). Similarly, we denote by
Csk,2+α (A) the space of all functions f on A whose k-th order derivatives Dxi Dyj Dtl f,
i + j + l = k, exist and belong to the space Cs2+α (A), as defined above. Both spaces,
equipped with the norms
X
||g||Csk,α (A) =
||Dxi Dyj Dtl g||Csα (A)
i+j+l≤k
and
||f ||Csk,2+α (A) =
X
i+j+l≤k
||Dxi Dyj Dtl f ||Cs2+α (A) ,
are Banach spaces. The last norm is equivalent to the norm
||f ||Csk,2+α (A) =
X
i+j+l≤k+1
||Dxi Dyj Dtl f ||Csα (A) +
X
i+j+l=k+2
l≤k
||xDxi Dyj Dtl f ||Csα (A) .
We denote by Cs0,α (A) and Cs0,2+α (A) the spaces Csα (A) and Cs2+α (A) respectively.
The operator
L0 : Csk,2+α (A) → Csk,α (A)
defines a continuous linear map.
Denoting by S0 the half-space x ≥ 0 in R2 , by S the space S = S0 × [0, ∞), and
by ST the space S0 × [0, T ], for T > 0, we state now the main result of Part I:
I.1.1. Theorem (Existence and Uniqueness). Let k be a nonnegative integer and
let α be a number in 0 < α < 1. Assume that g ∈ Csk,α (S) and f 0 ∈ Csk,2+α (S0 ),
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
903
with both g and f 0 compactly suported in S and S0 respectively. Then, for any
ν > 0 and T > 0, the initial value problem
(
L0 f = g
in ST ,
f (·, 0) = f 0
on S0
admits a unique solution f ∈ Csk,2+α (ST ). Moreover
||f ||Csk,2+α (ST ) ≤ C(T ) ||f 0 ||Csk,2+α (S0 ) + ||g||Csk,α (S)
for some constant C(T ) depending only α, k, ν and T .
Theorem I.1.1 follows from the following two results:
I.1.2. Theorem (Existence and Uniqueness of Smooth Solutions). Assume that g
is a smooth function with compact support on S = S0 × [0, ∞), which vanishes at
t = 0. Then, for any ν > 0, there exists a unique smooth solution f of the initial
value problem
(
in S,
L0 f = g
f (·, 0) = 0
on S0 .
Moreover, for any T > 0 there exists a constant C(T ) depending only on ν and T
so that
||f ||C 0 (ST ) ≤ C(T ) ||g||C 0 (S) .
x0 Define the box of side r around a point P = y0 to be
t0
 

x ≥ 0, |x − x0 | ≤ r 
 x
Br (P ) = y  :
.
|y − y0 | ≤ r


t
t0 − r ≤ t ≤ t0
0
We let Br be the box around the point P = 0 .
1
I.1.3. Theorem (Schauder Estimate). Let k be a nonnegative integer and let 0 <
α < 1 and ν > 0. Then, for any r < 1 there exists a constant C depending on k,
α, ν and r so that
kf kCsk,2+α(Br ) ≤ C kf kCs◦(B1 ) + kL0 f kCsk,α (B1 )
for all C ∞ smooth functions f on B1 .
I.2. The metric. We begin by elaborating on the metric
ds2 =
dx2 + dy 2
.
2x
The nonzero components of gij where
ds2 = gij dxi dxj
are
1
.
2x
are
gxx = gyy =
The nonzero components of the inverse g ij
g xx = g yy = 2x .
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
904
P. DASKALOPOULOS AND R. HAMILTON
The area element a = µ dx dy with
µ=
p
det gij
is given by
µ=
The Laplace operator in general is
1 ∂
∆s f =
µ ∂xi
1
.
2x
ij ∂
µg
f
∂xj
and in our case this gives
∆s f = 2x(fxx + fyy ) .
The Christoffel symbols are given in general by
1
∂
∂
∂
Γ`ij = g k`
g
+
g
−
g
jk
ik
ij
2
∂xi
∂xj
∂xk
and in our case the nonzero Christoffel symbols are
1
1
1
Γxyy = + ,
Γyxy = −
.
Γxxx = − ,
2x
2x
2x
The shortest distance between two points is a geodesic, which in general satisfies
the equations
i
j
d2 xk
k dx dx
=0
+
Γ
ij
ds2
ds ds
where s is the arc length; in our case we get the system
 2
2
2
d x
1 dx
1 dy


+
= 0,
 2 −
ds
2x ds
2x ds
2


 d y − 1 dx dy = 0 .
ds2
x ds ds
A curve parametrized by arc length has speed one, so
" 2 #
2
1
dx
dy
=1.
+
2x
ds
ds
We can solve for dy/ds in terms of dx/ds and substitute in the equation for d2 x/ds2
to get an equation independent of y; then substituting v = dx/ds reduces this to a
first order equation which is separable after the substitution v = xp. This allows
us to solve explicitly for the geodesics.
I.2.1. Proposition. The cycloid curve
x = 1 − cos s,
y = s − sin s
is a geodesic parametrized by arc length s for the metric
dx2 + dy 2
2x
and all the other geodesics are obtained by translation y → y+b and dilation x → cx,
y → cy, or are horizontal lines.
ds2 =
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
905
Proof. The reader can verify that the cycloid solves the equations for a geodesic
by substitution. Translations leave the metric fixed, while dilations expand it by a
fixed factor, so the translates and dilates are also geodesics. Since there is one such
curve through each point with x > 0 in each direction, this is all of them. To see
this, pick the point on the cycloid arch for 0 < s < 2π with the desired slope, and
then translate and dilate it till it reaches the desired point. This works because
the slope is invariant under translation and dilation. The only exception is that
the horizontal slope is attained only at the ends s = 0 or 2π. This corresponds to
horizontal lines, which are also solutions of the geodesic equations, and correspond
to limits of dilations x → cx, y → cy as c → ∞.
Note that the boundary curve x = 0 is at a finite distance from the interior.
Along the horizontal lines dy = 0 and
dx
ds = √
2x
so
s=
√
2x
exactly; while along the catenoid
x = 1 − cos s =
so
s=
s4
s2
−
+ ...
2
24
√ x
+ ...
2x 1 +
24
and both are finite. By contrast along a vertical line dx = 0 and
dy
ds = √
2x
so the length of a vertical segment from ( xy10 ) to ( xy20 ) goes to infinity as x0 → 0.
When x0 is very small compared to |y1 − y2 | the shortest path is along a catenoid
which at first plunges fearlessly into the interior nearly horizontally at first until
x is comparable to |y1 − y2 |, then moves vertically up where vertical distances are
much shorter, and finally returns toward the boundary nearly horizontally. This
motivates our understanding of our Hölder norms; near the boundary the action is
essentially always horizontal. Diffusion over a distance s in a time t takes place on
scales where s2 ≈ t. Near the boundary most of the diffusion is horizontal on scales
where x ≈ s2 so x ≈ t. Here the drift from the term νfx becomes comparable to
the diffusion. We need ν > 0 so that the drift is toward the boundary; if we were
in the case ν < 0 we would need a boundary condition.
In Section I.2 we will prove the existence and uniqueness of Theorem I.1.2.
Sections I.4-I.9 will be devoted to the proof of the Schauder estimates in Theorem
I.1.3. In Section I.10 we will prove an extension lemma for Hölder spaces with
respect to the metric s. The proof of Theorem I.1.1 will be given in Section I.12.
I.3. Proof of Theorem I.1.2. Assume that g is a smooth function with compact
support on S = S0 × [0, ∞), with S0 denoting, as before, the half-space x ≥ 0 in
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
906
P. DASKALOPOULOS AND R. HAMILTON
R2 , which vanishes at t = 0. We will show the existence of a smooth solution of
the initial value problem
(
L0 f = g
in S,
f (·, 0) = 0
on S0
with
L0 f = ft − x (fxx + fyy ) − ν fx
such that
||f ||C 0 (S) ≤ C ||g||C 0 (S)
for some absolute constant C. For this purpose, we convert the equation L0 f = g
to an ordinary differential equation in x with a regular singular point at x = 0, by
applying the Fourier-Laplace transform in the variables (y, t), t ≥ 0. Indeed, it is
easy to see that f is a solution of the equation
ft − x (fxx + fyy ) − ν fx = g
if and only if its Fourier-Laplace transform in (y, t), defined as
Z ∞
Z ∞
e−tτ dt
f (x, y, t) e−iyξ dy
fe(x, ξ, τ ) =
t=0
y=−∞
for ξ a real number and τ a complex number with Re(τ ) > 0, satisfies the ordinary
differential equation
g
x fexx + ν fex − (xξ 2 + τ ) fe = −e
with
Z
g (x, ξ, τ ) =
e
∞
e−tτ dt
t=0
Z
∞
g(x, y, t) e−iyξ dy.
y=−∞
Since g is smooth and compactly supported in S, it is standard to show that the
function fe is well defined, smooth and decays rapidly as |ξ| → ∞ and |τ | → ∞
with Re(τ ) > 0, while it is compactly supported in x ∈ [0, ∞). The point x = 0 is
a regular singular point for the ordinary differential equation satisfied by fe. Hence,
for all real ξ and complex τ with Re(τ ) > 0, the equation has a unique smooth
solution fe(x, ξ, τ ) on the half-line x ≥ 0. Moreover, the solution fe will depend
smoothly on the parameters ξ and τ , making fe a smooth function with respect to
all variables (x, ξ, τ ), Re(τ ) > 0. We will show that for every x ≥ 0, the function
fe(x, ξ, τ ) decays rapidly as |ξ| → ∞, |τ | → +∞, Re(τ ) > 0, and hence its inverse
Fourier-Laplace transform
Z +i∞+
Z +∞
etτ dτ
fe(x, ξ, τ ) eiyξ dξ
f (x, y, t) = lim
→0
−i∞+
ξ=−∞
is well defined on S, is smooth and is nothing but the desired solution of the equation
L0 f = g.
Indeed, if we write fe = p + iq, ge = h + ik and xξ 2 + τ = ρ + iσ, the ordinary
differential equation satisfied by fe becomes equivalent to the system
(
xpxx + νpx − ρp + σq = h,
xqxx + νqx − ρq − σp = k.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
907
Then F = (p2 + q 2 )/2 satisfies the differential inequality
x Fxx + νFx − 2ρF ≥ ph + qk.
By Young’s inequality we have
2
C
C
p + q2
− (h2 + k 2 ) ≥ −ρF − A(ξ, τ )
ph + qk ≥ −ρ
2
ρ
ρ
with A(ξ, τ ) = supx≥0 (h2 + k 2 )(x, ξ, τ ). Hence, since ρ = xξ 2 + Re(τ ) ≥ Re(τ ) the
function
C
Fe = F −
A
(Re(τ ))2
satisfies the differential inequality
x Fexx + ν Fex − Re(τ ) Fe ≥ 0.
Moreover Fe is smooth and bounded on x ≥ 0, for all real ξ and complex τ with
Re(τ ) > 0. It follows from the maximum principle that Fe ≤ 0, for all x ≥ 0 which
gives us the bound
C
sup |e
g |(x, ξ, τ )
|fe|(x, ξ, τ ) ≤
Re(τ ) x≥0
g |(x, ξ, τ ) decays rapidly as
with C an absolute, positive constant. Since supx≥0 |e
|ξ| → ∞, |τ | → ∞ with Re(τ ) > 0, it follows from this estimate that the function
f given by
Z +i∞+
Z +∞
tτ
e dτ
fe(x, ξ, τ ) eiyξ dξ
f (x, y, t) = lim
→0
−i∞+
ξ=−∞
is well defined and therefore a smooth solution of the equation L0 f = g. It is also
easy to see that
||f ||C 0 (ST ) < ∞
for all T > 0. In addition, for any positive integer n, we have
Z +i∞+
Z +∞
etτ dτ
τ n fe(x, ξ, τ ) eiyξ dξ.
Dtn f (x, y, t) = lim
→0
−i∞+
ξ=−∞
Therefore, denoting by L the Laplace transform, we have
L(Dtn f )(x, y, τ ) = τ n L(f )(x, y, τ )
for all τ > 0. This immediately implies that
Dtn f (x, y, 0) = 0
for all positive integers n, making f a smooth function on S with f (·, ·, 0) = 0.
This answers the existence question. The uniqueness of bounded smooth solutions
as well as the estimate ||f ||C 0 (S) ≤ C ||g||C 0 (S) follows from the classical maximum
principle, shown next.
I.3.1. Theorem (Maximum Principle). Assume that g is a smooth function with
compact support on S which vanishes at t = 0 and that f 0 is a smooth compactly
supported function on S0 . Let f be a smooth solution to the initial value problem
(
in S,
L0 f = g
0
f (·, 0) = f
on S0
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
908
P. DASKALOPOULOS AND R. HAMILTON
such that
||f ||C 0 (ST ) < ∞
for all T > 0, with ST = S0 × [0, T ]. Then, for any T > 0 there exists a constant
C(T ) depending only on ν and T so that
||f ||C 0 (ST ) ≤ C(T ) ||f 0 ||C 0 (S0 ) + ||g||C 0 (S) .
Proof. Assume first that g ≡ 0. Fix a number T > 0 and for µ ∈ (0, 1) define the
barrier function
hµ (x, y, t) = M + µ(t + 1)[ 2ν(T + 1)2 + 2(T + 1)x + y 2 ]
with
M = ||f 0 ||C 0 (S0 ) .
It is easy to see that
(hµ )t − x ((hµ )xx + (hµ )yy ) − (νhµ )x ≥ 0
on ST .
For R > 0, we denote by BR the box {0 ≤ x ≤ R, |y| ≤ R}. We wish to show that
when R is sufficiently large, then
f ≤ hµ
in BR × [0, T ].
To see that f ≤ hµ at the lateral boundary of the cylinder BR × [0, T ] is straightforward. Wherein, since the function w = f − hµ satisfies
wt − x (wxx + wyy ) − νwx ≤ 0
on BR × [0, T ]
it follows from the classical maximum principle that f ≤ hµ in BR × [0, T ]. Notice
that, because we have x(wxx + wyy ) = 0 at x = 0 and the transport velocity ν
is positive, the maximum of w cannot occur at x = 0. Therefore no boundary
condition needs to be imposed at x = 0. Now letting R → ∞ and µ → 0, we
conclude that f (x, y, t) ≤ M on ST . By similar arguments f (x, y, t) ≥ −M on ST
and hence the desired estimate holds true.
In the case that g is not zero, the estimate can be proven by similar comparison
arguments. If the function g vanishes for t ≥ T0 , the constant C = C(T ) can be
taken to be independent of T .
I.4. Barrier functions. We need to construct barrier functions to estimate the
derivatives of a solution to the homogeneous equation in a neighborhood of a point
on the boundary. We imitate a familiar construction in the interior. If
ϕ=
1
1
+
,
t
1 − x2 − y 2
then for suitable constants C < ∞ and c > 0 we have the barrier inequality
∂ϕ
> ϕxx + ϕyy − Cϕ2 + c
∂t
as the reader can easily check. Note ϕ → ∞ if t → 0 or x2 + y 2 → 1. This barrier
has been used frequently to do interior estimates of diffusion-reaction equations for
diffusion in the Euclidean metric. For diffusion in our cycloidal metric we need a
similar barrier.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
909
I.4.1. Definition. We say ϕ satisfies the cycloidal barrier inequality with transport velocity ν if there exist suitable constants C < ∞ and c > 0 such that
ϕt > x(ϕxx + ϕyy ) + νϕx − Cxϕ2 + cϕ3/2 + c .
I.4.2. Definition. The barrier set Φγν for a given γ < 1 is the collection of all
functions ϕ > 0 which
(a) are defined and smooth in a relatively open subset U of the box B1 where U
contains the box Bγ , and
(b) satisfy the cycloidal barrier inequality with transport velocity ν, for some
suitable constants C < ∞ and c > 0 which may depend on ϕ.
Our objective in this section is to prove the following result.
I.4.3. Theorem. There exists a function ϕ in the barrier set Φγν for any ν > 0
and γ < 1 which is proper, in the sense that the sublevel sets {ϕ ≤ B} are compact
for each constant B < ∞.
We begin with the following observation.
I.4.4. Theorem. The barrier set Φγν is a cone, in the sense that
(a) if ϕ1 and ϕ2 belong to Φγν , so does ϕ1 + ϕ2 , and
(b) if ϕ belongs to Φγν and λ > 0 is a constant, then λϕ belongs to Φγν also.
Proof. If ϕ1 is defined in U1 and ϕ2 in U2 , then ϕ1 + ϕ2 is defined in U1 ∩ U2 ,
which is still a relatively open neighborhood of (0, 0, 1). The inequality follows by
adjusting the constants C and c using
√ 3/2
3/2
3/2
.
≤ 2 ϕ1 + ϕ2
(ϕ1 + ϕ2 )2 ≥ ϕ21 + ϕ22 and (ϕ1 + ϕ2 )
Scaling ϕ also scales C and c.
This allows us to construct ϕ as a sum of pieces which will restrict the domain
of ϕ respectively to {t > 0}, {|y| < 1 − bt} with b small, and {x < 1} by having
ϕ → ∞ as we approach the new boundary. This will make ϕ proper.
Now we find explicit barriers; first at t = 0.
I.4.5. Theorem. For any ν > 0 we can find a > 0 so that the function
1
ϕ=
t(x + at)
is in the barrier cone Φγν .
Proof. We need to choose C < ∞ and c > 0 so that
Cxϕ2 − νϕx > xϕxx + cϕ3/2 − ϕt + c
for our given ν > 0, where ϕyy = 0. Differentiating, we find we need
ν
2x
c
x + 2at
Cx
+
>
+ 3/2
+ 2
+ c.
t2 (x + at)2
t(x + at)2
t(x + at)3
t (x + at)2
t (x + at)3/2
Multiplying by t2 (x + at)2 , we find that we need
2xt
+ ct1/2 (x + at)1/2 + x + 2at + ct2 (x + at)2 .
Cx + νt >
x + at
Now we use 0 < t ≤ 1 and 0 ≤ x < 1 and
2
2xt
≤ x
x + at
a
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
910
P. DASKALOPOULOS AND R. HAMILTON
and
ct1/2 (x + at)1/2 ≤ c[x + (a + 1)t]
and
ct2 (x + at)2 ≤ 4ct
if a ≤ 1 to find that we only need a ≤ 1 and
2
Cx + νt > 1 + c +
x + [2a + c(5 + a)]t
a
which happens for all x > 0 and t > 0 if a ≤ 1 and
2
C >1+c+
and
ν > 2a + c(5 + a) .
a
Given ν > 0, choose a > 0 and c > 0 small enough to satisfy the second inequality,
and then choose C to satisfy the first. This proves ϕ ∈ Φγν for small a > 0.
I.4.6. Theorem. For any b > 0, the function
1
ϕ=
(y − bt)2
satisfies the barrier inequality on {0 < y − bt < 2}.
Proof. We need to find C < ∞ and c > 0 with
ϕt > x(ϕxx + ϕyy ) + νϕx − Cxϕ2 + cϕ3/2 + c .
Since ϕx = 0 and ϕxx = 0, it suffices to make
ϕt > cϕ3/2 + c
and
Cϕ2 > ϕyy .
Differentiating, we find we need
c
2b
>
+c
(y − bt)3
(y − bt)3
which happens if 0 < y − bt < 2 and c < 2b/9, and we need
6
C
>
(y − bt)4
(y − bt)4
which happens if C > 6.
I.4.7. Corollary. For any b > 0 the function
1
1
+
ϕ=
(1 + y − bt)2
(1 − y − bt)2
on {|y| < 1 − bt} also belongs to the barrier cone Φγν provided ν > 0 and b < 1 − γ.
Proof. Since the inequality defining the barrier cone is preserved under the translation y → 1 + y and the flip 1 + y → 1 − y, the result follows. Note that in the set
{|y| < 1 − bt} we have
0 < 1 + y − bt < 2
and
0 < 1 − y − bt < 2
as required by the previous Theorem. Also note that the set {|y| < 1−bt, 0 ≤ t ≤ 1}
contains {|y| < γ} and hence Bγ provided b < 1 − γ.
Finally we construct a barrier function at x → 1.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
911
I.4.8. Theorem. For any ν > 0 and γ < 1 the function
1+t
ϕ=
(1 − x2 )2
lies in the barrier cone Φγν .
Proof. Since ϕyy = 0, we need to find constants C < ∞ and c > 0 with
ϕt + Cxϕ2 > xϕxx + νϕx + cϕ3/2 + c .
Now when 0 ≤ x < 1 and 0 ≤ t ≤ 1 we have
1
ϕt =
(1 − x2 )2
and
x
xϕ2 ≥
(1 − x2 )4
and
4
ϕ3/2 ≤
(1 − x2 )3
and
4x(1 + t)
8x
ϕx =
≤
2
3
(1 − x )
(1 − x2 )3
and
ϕxx =
(4 + 20x2 )(1 + t)
48
≤
.
(1 − x2 )4
(1 − x2 )4
Hence we only need
Cx
48x
8νx
4c
1
+
>
+
+
+c .
2
2
2
4
2
4
2
3
(1 − x )
(1 − x )
(1 − x )
(1 − x )
(1 − x2 )3
Multiplying by (1 − x2 )4 , we need
(1 − x2 )2 + Cx > 48x + 8νx(1 − x2 ) + 4c(1 − x2 ) + c(1 − x2 )4 .
Since 0 ≤ x < 1, we have
8νx(1 − x2 ) ≤ 8νx
so it suffices to take C ≥ 49 + 8ν so that
Cx ≥ [48x + 8νx(1 − x2 )] + x
and to take c so small that
(1 − x2 )2 + x > 4c(1 − x2 ) + c(1 − x2 )4
which must happen on 0 ≤ x ≤ 1 for some c > 0 since the interval is compact and
(1 − x2 )2 + x > 0
on the entire interval. In fact if x ≤ 1/2
(1 − x2 )2 ≥ 9/16 ≥ 1/2
so
(1 − x2 )2 + x ≥ 1/2
on 0 ≤ x ≤ 1, while 1 − x2 ≤ 1, so it suffices to take c ≤ 1/10.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
912
P. DASKALOPOULOS AND R. HAMILTON
At last we can construct our proper barrier function
1
1
1+t
1
+
+
+
ϕ=
t(x + at) (1 + y − bt)2
(1 − y − bt)2
(1 − x2 )2
for suitably small a and b as demonstrated. It is clear that ϕ is proper since ϕ → ∞
if t → 0 or x → 1 or |y| → 1 − bt. Since b can be as small as we like, the domain of
ϕ still includes the box Bγ . This proves our Theorem I.4.3.
I.5. Local derivative estimates. Now we prove estimates on the derivatives fx
and fy of a smooth solution f of the equation for cycloidal diffusion with transport
velocity ν > 0
ft = x(fxx + fyy ) + νfx + g
for a forcing term g. We assume f is defined in the box B1 and that in the box we
have bounds on f, g, gx and gy , but this time we also need to assume a bound on
gyy . To see why, consider the evolution of the derivatives fx and fy , which is given
by
fxt = x(fxxx + fxyy ) + (ν + 1)fxx + fyy + gy
and
fyt = x(fyxx + fyyy ) + νfyx + gy .
Note that fy satisfies the equation for cycloidal diffusion with transport velocity ν
and forcing term gy ; while fx satisfies the equation for cycloidal diffusion with the
new transport velocity ν + 1 and the forcing term fyy + gy . We will estimate fy
from gy , and then applying this again we will estimate fyy from gyy , and use the
estimate on fyy as well as that on gx to estimate fx .
Consider first the function
Y = (A + f 2 )fy2
for a large constant A ≥ 7 we can choose later. The evolution of Y is
Yt = 2(A + f 2 )fy fyt + 2f fy2 ft
which becomes
Yt = 2(A + f 2 )fy [x(fyxx + fyyy ) + νfyx + gy ]
+ 2f fy2 [x(fxx + fyy ) + νfx + g] .
It is easy to compute that if A ≥ 7, |f | ≤ 1 and
Y = (A + f 2 )fy2 ,
then
Yxx + Yyy ≥ 2(A + f 2 )fy (fyxx + fyyy ) + 2f fy2 (fxx + fyy ) + fy4 .
Also
Yx = 2(A + f 2 )fy fyx + 2f fy2 fx .
This gives us the estimate
Yt ≤ x(Yxx + Yyy ) + νYx − xfy4 + 2(A + f 2 )fy gy + 2f fy2g .
Then as before we use |g| ≤ 1 and |gy | ≤ 1 and A + f 2 ≤ 2A to write
Yt ≤ x(Yxx + Yyy ) + νYx − xfy4 + 4A|fy | + 2fy2
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
913
which gives
Yt ≤ x(Yxx + Yyy ) + νYx − xfy4 + |fy |3 + KA3/2
for some constant K independent of A. Now
Y
Y
Y
≤ fy2 =
≤
2
2A
A+f
A
so
Yt ≤ x(Yxx + Yyy ) + (ν + 1)Yx − x
Y
2A
2
+
Y
A
3/2
+ KA3/2 .
Let Ye = Y /B. Then
B e 2 B 1/2 e 3/2 KA3/2
.
xY + 3/2 Y
+
Yet ≤ x(Yexx + Yeyy ) + ν Yex −
4A2
B
A
We want to make B/A2 large, and both B 1/2 /A3/2 and A3/2 /B small. Take B =
A5/2 . Then
A1/2 e 2
1
K
xY + 1/4 Ye 3/2 +
.
Yet ≤ x(Yexx + Yeyy ) + ν Yex −
4
A
A
Given any C < ∞ and c > 0, we can take A so large that
A1/2
≥C
4
and
1
≤c
A1/4
K
≤c
A
and
and get
Yet ≤ x(Yexx + Yeyy ) + ν Yex − CxYe 2 + c Ye 3/2 + c.
Now our proper barrier function ϕ gives a bound on Ye , which in turn gives a
bound on fy in the smaller box Bγ for any γ < 1. The bound Ye ≤ ϕ follows from
the maximum principle. Notice that in applying the maximum principle at the
boundary we need to have ν ≥ 0, because if Ye ≤ ϕ in the interior and Ye = ϕ at a
boundary point, then Yex ≤ ϕx at the boundary point, and we need this inequality
to enter with the correct sign. However if we try to prove the theorem for ν = 0,
we have trouble with the barrier function inequality for ϕ at the boundary. This
proves the following.
I.5.1. Theorem. If f is smooth and satisfies the cycloidal diffusion equation with
transport velocity ν > 0
ft = x(fxx + fyy ) + νfx + g
with forcing term g, and if
|f | ≤ 1
and
|g| ≤ 1
and
|gy | ≤ 1
on the box B1 = {0 ≤ x ≤ 1, −1 ≤ y ≤ 1, 0 ≤ t ≤ 1}, then for any γ < 1
|fy | ≤ C
on the box
Bγ = {0 ≤ x ≤ γ, −γ ≤ y ≤ γ, 1 − γ ≤ t ≤ 1}
for some constant C depending on ν > 0 and γ < 1 but independent of f and g.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
914
P. DASKALOPOULOS AND R. HAMILTON
By dilation we get the following:
I.5.2. Corollary. If f is smooth and satisfies the cycloidal diffusion equation with
transport velocity ν > 0
ft = x(fxx + fyy ) + νfx + g
with forcing term g, and if
|f | ≤ B
|g| ≤ B
and
and
|gy | ≤ B/r
on the box Br , then
|fy | ≤ C B/r
on the box Bγr for any γ < 1.
Since fy satisfies the cycloidal diffusion equation with transport velocity ν > 0
fyt = x(fyxx + fyyy ) + νfyx + gy
and forcing term gy , we can apply this result again to get a bound on fyy .
I.5.3. Corollary. If in addition
|gyy | ≤ B/r2
on the box Br/2 , then
|fyy | ≤ CB/r2
on the box Br/4 .
At last we are prepared to tackle the estimate for fx . To simplify the discussion
we will work on B1 and scale later. Suppose then that
ft = x(fxx + fyy ) + νfx + g
on B1 and that we have bounds
|f | ≤ 1
and
|fyy | ≤ 1
and
|g| ≤ 1
and
|gx | ≤ 1
on B1 . Consider the quantity
X = (A + f 2 )fx2
where A is a large constant we can choose later. The evolution of X is
Xt = 2(A + f 2 )fx fxt + 2f fx2 ft
and since
fxt = x(fxxx + fxyy ) + (ν + 1)fxx + fyy + gx
we find
Xt = 2(A + f 2 )fx [x(fxxx + fxyy ) + (ν + 1)fxx + fyy + gx ]
+ 2f fx2 [x(fxx + fyy ) + νfx + g] .
On the other hand, by direct computation if A ≥ 7 and |f | ≤ 1 we still have
Xxx + Xyy ≥ 2(A + f 2 )fx (fxxx + fxyy ) + 2f fx2 (fxx + fyy ) + fx4 .
Also
Xx = 2(A + f 2 )fx fxx + 2f fx3 .
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
915
Combining these gives
Xt ≤ x(Xxx + Xyy ) + (ν + 1)Xx − xfx4 − 2f fx3
+ 2(A + f 2 )fx (fyy + gx ) + 2f fx2 g .
When A ≥ 7 and |f | ≤ 1 we have A + f 2 ≤ 2A, and we also assume |g| ≤ 1 and
|gy | ≤ 1 and |fyy | ≤ 1. This gives
Xt ≤ x(Xxx + Xyy ) + (ν + 1)Xx − xfx4 + 2|fx |3 + 4A|fx | + 2fx2
and so with some constant K independent of A we have
Xt ≤ x(Xxx + Xyy ) + (ν + 1)Xx − xfx4 + 3|fx |3 + KA3/2 .
Now
X
X
X
≤ fx2 =
≤
2A
A + f2
A
so
Xt ≤ x(Xxx + Xyy ) + (ν + 1)Xx − x
X
2A
2
+3
X
A
3/2
+ KA3/2 .
e = X/B. Then
Let X
1/2
3/2
exx + X
eyy ) + (ν + 1)X
ex − B xX
e 2 + 3B X
e 3/2 + KA
et ≤ x(X
.
X
2
4A
B
A3/2
We want to make B/A2 large, and both B 1/2 /A3/2 and A3/2 /B small. Take B =
A5/2 . Then
1/2
e2 + 3 X
exx + X
eyy ) + (ν + 1)X
ex − A xX
e 3/2 + K .
et ≤ x(X
X
4
A
A1/4
Now given any C < ∞ and c > 0, we can take A so large that
A1/2
≥C
4
and
3
A1/4
≤c
and
K
≤c
A
and get
et ≤ x(X
exx + X
eyy ) + (ν + 1)X
ex − CxX
e 2 + cX
e 3/2 + c .
X
e ≤ ϕ for our barrier function ϕ. Since
Now the maximum principle shows that X
ν > 0 we also have ν + 1 > 0; therefore if the maximum occurs at the boundary,
ex ≤ ϕx comes in with the right sign. This establishes the following
the term from X
result.
I.5.4. Theorem. Suppose f is smooth and satisfies the cycloidal diffusion equation
with transport velocity ν > 0
ft = x(fxx + fyy ) + νfx + g
with forcing term g on the box
B1 = {0 ≤ x ≤ 1, −1 ≤ y ≤ 1, 0 ≤ t ≤ 1} .
If
|f | ≤ 1
and
|fyy | ≤ 1
and
|g| ≤ 1
on B1 , then
|fx | ≤ C
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
and
|gx | ≤ 1
916
P. DASKALOPOULOS AND R. HAMILTON
on the box
Bγ = {0 ≤ x ≤ γ, −γ ≤ y ≤ γ, 1 − γ ≤ t ≤ 1}
with a constant C which may depend on ν > 0 and γ < 1 but is independent of f
and g.
By dilation we get the following.
I.5.5. Corollary. If f and g satisfy the equation on the box Br and
|f | ≤ B
and
|fyy | ≤ B/r2
|g| ≤ B
and
and
|gx | ≤ B/r
on Br , then
|fx | ≤ C B/r
on the box Bγr .
I.5.6. Corollary. If f and g satisfy the equation on the box Br and if
|f | ≤ B
and
|g| ≤ B
and
|gx | ≤ B/r
and
|gy | ≤ B/r
and
|gyy | ≤ B/r2
on the box Br , then for any θ < 1
|fx | ≤ CB/r
and
|fy | ≤ CB/r
on the box Bθr .
Proof. The bound on fy on Bγr follows first from the equation for f and the yderivative estimate, which uses bounds on g and gy on Br . The bound on fyy on
Bγ 2 r follows next from the equation for fy and the y-derivative estimate, which has
gy as forcing term and hence uses bounds on gyy on Bγr also. Now the bound on fx
on Bγ 3 r follows from the equation for f and the x-derivative estimate, which uses
the bound on fyy on Bγ 2 r found above and the bound on gx on Bγ 2 r . Take γ = θ1/3
so θ = γ 3 . Continuing in this manner we can bound any derivative of f we wish in
any smaller box, using only bounds on f , and bounds on g and the derivatives of
g. If f solves the homogeneous equation, we don’t need to keep track.
I.5.7. Corollary. If f is smooth and solves the homogeneous cycloidal diffusion
equation with transport velocity ν > 0
ft = x(fxx + fyy ) + νfx ,
then for any i and j and any γ < 1 we can find a constant C depending on i and j
and ν > 0 and γ < 1, but not on f or on r, so that if |f | ≤ B on the box Br , then
|Dxi Dyj f | ≤ CB/ri+j
on the box Bγr .
Proof. Note that even if f satisfies the homogeneous equation, then so does fy but
fx does not. Thus even to prove the result for the homogeneous equation we need
to consider the inhomogeneous case g 6= 0 and bound derivatives of f in terms of
derivatives of g. Since
fxt = x(fxxx + fxyy ) + (ν + 1)fxx + (fyy + gx )
and
fyt = x(fyxx + fyyy ) + νfy + gy
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
917
we bound Dxi Dyj f first by induction on the number j of y-derivatives for fixed i,
and then by induction on the number i of x-derivatives.
I.6. Polynomial approximation. In order to make Schauder estimates we use a
technique of approximation by a polynomial developed in [S1], [S2], [C], [W1] and
[W2]. We will first prove a polynomial approximation theorem for the cycloidal
diffusion operator with transport velocity ν > 0
L0 f = ft − x(fxx + fyy ) − νfx
at points on the boundary. The differences will be that now time scales like space
instead of like space squared, and we only take a polynomial of degree one in space
and time instead of degree two in space, because at the boundary the diffusion goes
to zero. We now let kf kr be the maximum of f on the box Br .
I.6.1. Cycloidal Polynomial Approximation Theorem. There exists a constant C with the following property. For every smooth function f on the box Bs we
can find a polynomial p of degree one in space and time so that for every r ≤ s
r 2
kf ks + skL0 f ks .
kf − pkr ≤ C
s
Proof. By scaling it suffices to prove the result for s = 1. Suppose then that f is
smooth on B1 , and we shall show
kf − pkr ≤ C r2 kf k1 + kL0 f k1 .
Again choose a bump function ψ on the set
S = {0 ≤ x < ∞, −∞ < y < ∞, −∞ < t ≤ 1}
so that ψ = 0 outside B1 and ψ = 1 on B1/2 . Let L0 f = g, and choose h to be the
unique bounded smooth solution on S of the equation
L0 h = ψg.
Let k = f − h, and let p be the Taylor polynomial of k
 
 
 
 
 
x
0
0
0
0
p y  = k 0 + kx 0 x + ky 0 y + kt 0 (t − 1)
t
1
1
1
1
x
0
of degree 1 in space and time at the point 0 . For any point y in Br the
t
1
remainder k − p is given by terms involving derivatives of k at another point in Br
of higher order times the corresponding monomial.
Now k satisfies the homogeneous equation on B1/2 , since
L0 k = L0 f − L0 h = g − ψg
and ψ = 1 on B1/2 . Therefore we can bound any derivative of k onB1/4
by a
0
constant times the bound kkk1 on k. The derivatives of k at the point 0 give a
1
bound on the Taylor polynomial p,
kpk1 ≤ Ckkk1 .
To bound the remainder k − p on Br , first suppose r ≤ 1/4 so we have bounds on
the derivatives of k on Br . The terms in the remainder formula are monomials like
x2 and xy and x(t − 1) and (t − 1)2 of higher degree than one in space and time.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
918
P. DASKALOPOULOS AND R. HAMILTON
Since 0 ≤ x ≤ r and |y| ≤ r and 0 ≤ 1 − t ≤ r on Br , all these monomials are
bounded by r2 . This gives the remainder estimate
kk − pkr ≤ Cr2 kkk1
when r ≤ 1/4. Of course if r ≥ 1/4 it is much easier, r2 doesn’t matter and
kk − pkr ≤ kk − pk1 ≤ kkk1 + kpk1
and we already have kpk1 ≤ Ckkk1 .
We can estimate khk1 from the equation
L0 h = ht − x(hxx + hyy ) − νhx = ψg
using the maximum principle, since the maximum of |ψg| everywhere is no more
than that of g on B1 , namely kgk1 . Since h = 0 for t ≤ 0 because ψ = 0 for t ≤ 0,
and the maximum of h grows at most at a rate kgk1 for 0 ≤ t ≤ 1, we get
khk1 ≤ kgk1 .
Since k = f − h
kkk1 ≤ kf k1 + khk1 ≤ kf k1 + kgk1 .
Now
f − p = (f − k) + (k − p) = h + (k − p)
so
kf − pkr ≤ khkr + kk − pkr
and
khkr ≤ khk1 ≤ kgk1
and
kk − pkr ≤ Cr2 kkk1 ≤ Cr2 (kf k1 + kgk1 )
which makes
kf − pkr ≤ C(r2 kf k1 + kgk1 )
which is the estimate we desire since g = L0 f.
I.7. Taylor remainder estimates. Now we derive Schauder estimates along the
lines of Caffarelli, Safonov and Wang. Let
L0 f = ft − x(fxx + fyy ) − νfx
be our operator this time, and let kf kr be the supremum of f over the box Br .
I.7.1. Cycloidal Schauder Estimate. For each ν > 0 and each α in 0 < α < 1
there exists a constant S with the following
0 property. If f is a smooth function on
the box B1 whose Taylor polynomial at 0 of degree 1 in space and time is zero,
1
then
kf kr
kL0 f kr
sup
.
≤
S
kf
k
+
sup
1
1+α
rα
0<r≤1 r
0<r≤1
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
919
Proof. Since f is smooth and its Taylor polynomial of degree 1 vanishes,
kf kr
=0;
r1+α
0
likewise since the constant value of L0 f at 0 is zero,
lim
r→0
1
kL0 f kr
=0
rα
also. Thus both suprema are attained, although probably not at the same r. Suppose now that we let r be the exact value of u where
lim
r→0
sup
0<u≤1
kf kr
kf ku
= 1+α
1+α
u
r
is attained.
First note that if we know r ≥ c for some constant c > 0, we can take S = 1/c1+α
since kf kr ≤ kf k1. Thus there is no problem unless r is very small. In this case we
shall choose q much smaller than r and s much larger, of course with s ≤ 1 which
is why we need r small. We shall estimate S by making q/r and r/s both small
enough (by amounts to be determined later).
Let S be the best constant that works in the estimate, so that if (for ease)
Q = kf k1 + sup
0<u≤1
kL0 f ku
,
uα
then
kf kr
= SQ .
r1+α
We shall estimate S from this.
By the cycloidal polynomial approximation theorem choose a polynomial p of
degree 1 in space and time so that for all q ≤ s we have
q 2
kf − pkq ≤ C
kf ks + skL0 f ks
s
and since in particular r ≤ s we also have
r 2
kf ks + skL0 f ks .
kf − pkr ≤ C
s
Since p is a polynomial of degree 1
r
kpkr ≤ C
kpkq
q
because we can estimate the coefficients of p by kpkq /q, and we can estimate kpkr
by the coefficients times r. We also use
kf kr ≤ kf − pkr + kpkr
and
kpkq ≤ kf − pkq + kf kq
to get
h
i
r
kf − pkq + kf kq .
kf kr ≤ kf − pkr + C
q
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
920
P. DASKALOPOULOS AND R. HAMILTON
Then our approximation estimates on f − p give
r 2
rs
r
kf kr ≤ C kf kq +
kf ks + kL0 f ||s .
q
s
q
From the definition of S
kf kr = Sr1+α Q
while
kf kq ≤ Sq 1+α Q
and
kf ks ≤ Ss1+α Q,
and from the definition of Q
kL0 f ks ≤ sα Q .
Plug all this in and divide by r1+α Q to get
s1+α
q α r 1−α
S≤C
S+C α .
+
r
s
qr
If q/r and r/s are small enough compared to the constant C, we get
1
q α r 1−α
≤
+
C
r
s
2
and this gives a bound on S as desired.
Next we show how to remove the restriction on the Taylor polynomial vanishing,
so that we get an estimate for a general function on the Taylor remainder. For a
smooth function f on the box B1 we let T1 f denote
0 the Taylor polynomial of f of
degree 1 in both space and time at the point 0 ,
1
 
 
 
 
 
x
0
0
0
0
T1 f y  = f 0 + fx 0 x + fy 0 y + ft 0 (t − 1),
t
1
1
1
1
and we let
R1 f = f − T 1 f
denote the remainder. Likewise for a smooth function g on B1 we let T0 g denote
0
the Taylor polynomial of degree 0 in both space and time at the point 0 ,
1
 
 
x
0
T0 g y  = g 0
t
1
(so that T0 g is a constant function), and we let
R0 g = g − T0 g
denote the remainder.
I.7.2. Corollary. For any smooth function f on the box B1
kR1 f ||r
kR0 L0 f kr
.
sup
≤
S
kR
f
k
+
sup
1
1
r1+α
rα
0<r≤1
0<r≤1
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
921
Proof. Apply Theorem I.7.1 to the remainder function R1 f , whose Taylor polynomial T1 R1 f = 0.
Recall that now for cycloidal diffusion
kf kr = kf kC 0(Br ) = sup |f (P )|
P ∈Br
and
kf kCsβ (Br ) = kf kC 0 (Br ) +
with
sup
P1 ,P2 ∈Br
|f (P1 ) − f (P2 )|
s[P1 , P2 ]β

  
x1
x2
p
x
x1






y1 , y2
, 2
+ |t1 − t2 |
s
=s
y1
y2
t1
t2
and
|x1 − x2 | + |y1 − y2 |
x
x1
p
, 2
≤C√
√
y1
y2
x1 + x2 + |y1 − y2 |
0
for constants c > 0 and C < ∞. When one of the points is P = 0 , we have the
1
simpler comparison
|x1 − x2 | + |y1 − y2 |
p
≤s
c√
√
x1 + x2 + |y1 − y2 |
   
x
0
p
p
p
p
p
c
|x| + |y| + |t − 1| ≤ s y  , 0 ≤ C
|x| + |y| + |t − 1| ,
t
1
x
so for points y in Br
t
   
x
0
√
s y  , 0 ≤ C r
1
t
√ 2α
for constants c > 0 and C < ∞. Since ( r) = rα , our estimates in terms of rα
produce Hölder estimates of exponent β = 2α. For example, for all smooth g on B1
p
sup
0<r≤1
kR0 gkr
≤ CkgkCsβ (B1 ) .
rα
Now we can
bound the Taylor polynomial T1 f , and hence the derivatives of f
0
at the point 0 of degree 1 in both space and time.
1
I.7.3. Theorem. For every smooth function f on the box B1 and every β in 0 <
β<1
kT1 f kC 0 (B1 ) ≤ C kf kC 0 (B1 ) + kL0 f kCsβ (B1 ) .
Proof. From Corollary I.7.2, for all r in 0 < r ≤ 1 and with α = β/2
kR1 f kr ≤ Sr1+α kR1 f k1 + kL0 f kCsβ (B1 ) .
Since T1 f is a polynomial of degree 1 in both space and time
C
kT1 f k1 ≤ kT1 f kr
r
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
922
P. DASKALOPOULOS AND R. HAMILTON
for all r > 0. Then
kT1 f kr ≤ kf kr + kR1 f kr
and kf kr ≤ kf k1 , and
kR1 f k1 ≤ kf k1 + kT1 f k1 .
Combining these gives
C
kf k1 + Crα kL0 f kCsβ (B1 ) .
r
< 1/2, and the bound on kT1 f kC 0 (B1 ) = kT1 f k1
kT1 f k1 ≤ Crα kT1 f k1 +
Choose r so small that Crα
follows.
I.7.4. Corollary. If i + j + k ≤ 1, then
 
0 i j k
Dx Dy Dt f 0 ≤ C kf kC 0 (B ) + kL0 f k β
.
1
Cs (B1 )
1 I.7.5. Corollary. We also have
sup
0<r≤1
kR1 f kr
0
≤
C
kf
k
+
kL
f
k
β
0
C (B1 )
Cs (B1 )
r1+α
for all smooth f on B1 , with 0 < β = 2α < 1.
Proof. We use the bound in Corollary I.7.4 and
kR1 f k1 ≤ kf k1 + kT1 f k1
and the bound in Theorem I.7.3 on kT1 f k1 .
Next we show that we can replace the norm kL0 f kCsβ (B1 ) in Theorem I.7.3 and
Corollary I.7.4 by the seminorm kL0 f kHsβ (B1 ) .
I.7.6. Theorem. For every smooth function f on the box B1 and every α in 0 <
α<1
kT1 f kC 0 (B1 ) ≤ C kf kC 0 (B1 ) + kL0 f kHsβ (B1 ) .
Proof. From Corollary I.7.2 since
sup
0<r≤1
we have
kR0 L0 f kr
≤ CkL0 f kHsβ (B1 )
rα
kR1 f kC 0 (Br ) ≤ Cr1+α kR1 f kC 0 (B1 ) + kL0 f kHsβ (B1 ) .
Since T1 f is a polynomial of degree 1 in both space and time
C
kT1 f kC 0 (B1 ) ≤ kT1 f kC 0 (Br )
r
for all r > 0. Then
kT1 f kC 0 (Br ) ≤ kf kC 0 (Br ) + kR1 f kC 0 (Br )
and
kf kC 0 (Br ) ≤ kf kC 0 (B1 )
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
923
and
kR1 f kC 0 (B1 ) ≤ kf kC 0 (B1 ) + kT1 f kC 0 (B1 ) .
Combining these gives
kT1 f kC 0 (B1 ) ≤ Crα kT1 f kC 0 (B1 ) +
C
kf kC 0(B1 ) + Crα kL0 f kHsβ (B1 ) .
r
Choose r so small that Crα < 1/2, and the bound on T1 f follows.
I.7.7. Corollary. If i + j + k ≤ 1, then
 
0 i j k
Dx Dy Dt f 0 ≤ C kf kC 0 (B ) + kL0 f k β
.
1
Hs (B1 )
1 I.7.8. Corollary. We also have
kR1 f kC 0 (Br ) ≤ Cr1+α kf kC 0 (B1 ) + kL0 f kHsβ (B1 )
for all r in 0 ≤ r ≤ 1.
Proof. We use the bound in Corollary I.7.2 and
kR1 f kC 0 (B1 ) ≤ kf kC 0 (B1 ) + kT1 f kC 0 (B1 )
and the bound on T1 f from Theorem I.7.6.
I.8. Schauder estimates in the interior. Given a point P =
x0 the parabolic cylinder Cr (P ) of radius r around P to be the set
 

 x
2
2
2
|x − x0 | + |y − y0 | ≤ r
.
Cr (P ) = y  :
t0 − r 2 ≤ t ≤ t 0


t
Then
y0
t0
we define
 
0
Cr = Cr 0 .
1
If P ∈ C1/2 , then C1/2 (P ) ⊆ C1 . If
I = (i, j, k)
is a multi-index, we let
DI f = Dxi Dyj Dtk f .
We review the classical Schauder estimates for the heat operator
Hf = ft − (fxx + fyy ).
I.8.1. Theorem. For any r < 1 there exists a constant C < ∞ depending on r
with the following property. If f is any smooth function on the cylinder C1 , then
kf kC 2+α(Cr ) ≤ C kf kC 0 (C1 ) + kHf kC α(C1 ) .
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
924
P. DASKALOPOULOS AND R. HAMILTON
We first prove interior Schauder estimates for the cycloidal diffusion operator
with transport velocity ν
L0 f = ft − x(fxx + fyy ) − νfx
in a small cylinder around the interior point Q = {x = 1, y = 0, t = 1}. Since
Hf = ft − (fxx + fyy )
we have
Hf − L0 f = (x − 1)(fxx + fyy ) + νfx .
On the cylinder Cλ (Q) with λ < 1
kHf − L0 f kC α (Cλ (Q)) ≤ Cλkf kC 2+α (Cλ (Q)) + Ckf kC 1+α (Cλ (Q))
where C depends on ν but not λ. We have
kf kC 2+α(C1/4 (Q)) ≤ C kf kC 0(C1 (Q)) + kHf kC α(C1 (Q))
by translating Theorem I.8.1, with a constant independent of λ. If f has support
in Cλ , so f is defined for all x and y and t ≤ 1 and vanishes if |x − 1| ≥ λ or |y| ≥ λ
or t ≤ 1 − λ2 , then for all µ ≥ λ any norm of f on Cµ is the same as its norm on
Cλ . Thus
kf kC 2+α (Cλ (Q)) ≤ C kf kC 0 (Cλ (Q)) + kHf kC α(Cλ (Q))
with a constant independent of λ. We can estimate
kHf kC α(Cλ (Q)) ≤ kL0 f kC α (Cλ (Q)) + kHf − L0 f kC α (Cλ (Q)) .
This gives
kf kC 2+α(Cλ (Q)) ≤ Cλkf kC 2+α (Cλ (Q)) + Ckf kC 1+α (Cλ (Q)) + CkL0 f kC α (Cλ (Q)) .
Choose λ so small that Cλ ≤ 1/2; then we get the following result:
I.8.2. Theorem. There exists a number λ > 0 and a constant C with the following
1
property. For every function f with support in the cylinder Cλ (Q) with Q = 0
1
we have
kf kC 2+α(Cλ (Q)) ≤ C kf kC 1+α(Cλ (Q)) + kL0 f kC α (Cλ (Q)) .
We can improve the lower order norm.
I.8.3. Corollary. We also have
kf kC 2+α(Cλ (Q)) ≤ C kf kC 0 (Cλ (Q)) + kL0 f kC α (Cλ (Q))
.
Proof. For any ε > 0 we can find a constant Cε so that if f has support in Cλ (Q),
kf kC 1+α(Cλ (Q)) ≤ εkf kC 2+α(Cλ (Q)) + Cε kf ||C 0 (Cλ(Q))
by interpolation theory. If Cε ≤ 1/2, we can get the desired bound.
We also observe that it suffices to use the semi-norm H α on L0 f .
I.8.4. Corollary. We also have
kf kC 2+α (Cλ (Q)) ≤ C kf kC 0 (Cλ (Q)) + kL0 f kH α (Cλ (Q))
.
Proof. Since g = L0 f also has support in Cλ (Q)
kgkC 0(Cλ (Q)) ≤ CkgkH α (Cλ (Q))
comparing the point where |g| is a maximum to a point where g = 0.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
925
We can also drop the restriction that f has support in Cλ (Q).
I.8.5. Theorem. There exists λ > 0 such that for any µ < λ and any smooth
function f on Cλ (Q) without restriction on its support
kf kC 2+α (Cµ (Q)) ≤ C kf kC 0 (Cλ (Q)) + kL0 f kH α (Cλ (Q)) .
Proof. Choose ν, ρ, σ and θ with
µ<ν<ρ<σ<λ<θ
and choose bump functions ϕ and ψ so that
ϕ=1
on
ϕ=0
off
and
ψ=1
ψ=0
on
off
Cν ,
Cρ
Cσ ,
Cλ .
Solve for the unique bounded function h with
L0 h = ϕL0 f .
By the maximum principle
khkC 0 (Cθ (Q)) ≤ CkϕL0 f kC 0 (Cρ (Q)) ≤ CkL0 f kH α (Cρ (Q))
because ϕL0 f has compact support in Cρ (Q). Since L0 h = 0 off of Cρ , the interior
derivative estimates on solutions of the homogeneous equation give
khkC 1+α(Cλ (Q)−Cσ (Q)) ≤ khkC 2 (Cλ (Q)−Cσ (Q))
≤ khkC 0 (Cθ (Q)−Cρ (Q)) ≤ CkL0 f kH α (Cρ (Q))
since we can cover Cλ (Q) − Cσ (Q) with a finite number of balls in slightly larger
balls in Cθ (Q) − Cρ (Q). Since the commutator [L0 , ψ] has support in Cλ (Q) − Cσ (Q)
and degree 1 in space and zero in time
k[L0 , ψ]hkC α (Cλ (Q)) ≤ CkhkC 1+α (Cλ (Q)−Cσ (Q)) ≤ CkL0 f kH α (Cρ (Q)) .
Now
L0 ψh = ψL0 h + [L0 , ψ]h
and ψϕ = ϕ so
L0 ψh = ϕL0 f + [L0 , ψ]h .
Since ψh has compact support in Cλ (Q)
kψhkC 2+α (Cλ (Q)) ≤ C kψhkC 0 (Cλ (Q)) + kL0 ψhkH α (Cλ (Q))
by Corollary I.8.4. Therefore, since
kL0 ψhkH α (Cλ (Q)) ≤ kϕL0 f kH α (Cλ (Q)) + k[L0 , ψ]hkH α (Cλ (Q)) ≤ CkL0 f kH α (Cρ (Q))
and
kψhkC 0 (Cλ (Q)) ≤ khkC 0 (Cλ (Q)) ≤ CkL0 f kH α (Cρ (Q)) ,
we get
khkC 2+α(Cµ (Q)) ≤ kψhkC 2+α(Cλ (Q)) ≤ CkL0 f kH α (Cρ (Q)) .
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
926
P. DASKALOPOULOS AND R. HAMILTON
Next let k = f − h; then
L0 k = L0 f − L0 h = (1 − ϕ)L0 f
so Lk = 0 on Cν . By the interior derivative estimates on solutions of the homogeneous equation
kkkC 2+α(Cµ (Q)) ≤ CkkkC 3 (Cµ (Q)) ≤ CkkkC 0 (Cν (Q)) ≤ CkkkC 0 (Cλ (Q)) .
Now k = f − h so
kkkC 0 (Cλ (Q)) ≤ kf kC 0(Cλ (Q)) + khkC 0 (Cλ (Q))
and once again
khkC 0 (Cλ (Q)) ≤ CkL0 f kH α (Cρ (Q))
so we get
kkkC 2+α(Cµ (Q)) ≤ C kf kC 0(Cλ (Q)) + kL0 f kH α (Cρ (Q))
.
Now f = h + k so
kf kC 2+α(Cµ (Q)) ≤ khkC 2+α(Cµ (Q)) + kkkC 2+α (Cµ (Q))
≤ C kf kC 0(Cλ (Q)) + kL0 f kH α (Cρ (Q))
as desired.
On the cylinder Cλ (Q) the metric
dx2 + dy 2
2x
is equivalent to the Euclidean metric since |x − 1| ≤ λ and λ is small. This gives
the following restatement, replacing α by β and H α by Hsβ in Theorem I.8.5.
ds2 =
I.8.6. Corollary. We also have
kf kCs2+β (Cµ (Q)) ≤ C kf kC 0 (Cλ (Q)) + kL0 f kHsβ (Cλ (Q)) .
For dilation purposes we introduce the semi-norm Hs2+β on a set A:
kf kHs2+β (A) = kxfxx kHsβ (A) + kxfxy kHsβ (A) + kxfyy kHsβ (A)
+ kfx kHsβ (A) + kfy kHsβ (A) + kft kHsβ (A) .
Clearly Hs2+β is weaker than Cs2+β , so
kf kHs2+β (Cµ (Q)) ≤ C kf kC 0 (Cλ (Q)) + kL0 f kHsβ (Cµ (Q)) .
Each of these norms behaves well under dilation. If we dilate space and time by a
constant factor r, then L0 f dilates by r, the C 0 norm is unchanged, the Hsβ norm
dilates by sβ = rα with α = β/2, and the Hs2+β norm dilates by rsβ = r1+α .
Let Qr be the point
Qr = {x = r, y = 0, t = 1}
and let
(
Aλr (Qr ) =
(x − r)2 + y 2 ≤ λ2 r2
1 − λ2 r ≤ t ≤ 1
)
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
927
be the cylinder obtained by scaling the cylinder
(
)
(x − 1)2 + y 2 ≤ λ2
Cλ (Q) =
1 − λ2 ≤ t ≤ 1
by x → rx, y → ry, (1 − t) → r(1 − t).
I.8.7. Corollary. There exists a λ > 0 such that for every µ < λ and every β in
0 < β < 1 we can find a constant C with the following property. For every r > 0
and every smooth function f on the cylinder Aλr (Qr )
1
kf kHs2+β (Aµr (Qr )) ≤ C
kf kC 0 (Aλr (Qr )) + kL0 f kHsβ (Aλr (Qr ))
r1+α
with α = β/2.
From this Schauder estimate we can work backwards to get a Taylor remainder
Qr
f denote the Taylor polynomial of f of degree 2 in space and 1
estimate. Let T2,1
in time at the point Qr , and let
Qr
Qr
R2,1
f = f − T2,1
f
Qr
f in
be the Taylor remainder at Qr . By the remainder formula we can express R2,1
terms of the differences of derivatives fxx , fxy , fyy , ft between Qr and the near by
points, so that, as we see by dilating from Aµ (Q),
Qr
f kC 0 (Aµr (Q)) ≤ Cr1+α kf kHs2+β (Aµr (Qr )) .
kR2,1
Combining this with the previous estimate gives this corollary.
I.8.8. Corollary. We also have (with α = β/2)
Qr
f kC 0 (Aµr (Qr )) ≤ C kf kC 0 (Aλr (Qr )) + r1+α kL0 f kHsβ (Aλr (Qr )) .
kR2,1
I.9. Schauder estimates near the boundary. We can obtain Schauder estimates near
the boundary comparing the second derivatives fxx , fxy , fyy at
apoint
r
0
P = 0 on the boundary with the second derivatives at a point Qr = 0 near
1
1
the boundary, by comparing the Taylor remainder estimates near P and near Qr .
Let T1P f denote the Taylor polynomial of f at P of degree 1 in both space and
Qr
f denote the Taylor polynomial of f at Qr of degree 2 in space
time, and let T2,1
and 1 in time, and consider the remainders
R1P f = f − T1P f
and
Qr
Qr
R2,1
f = f − T2,1
f .
For λ small the cylinder Aλr (Qr ) is entirely contained in the box B2r (P ). Our
remainder estimate at the boundary gives
kR1P f kC 0 (B2r (P )) ≤ Cr1+α kf kC 0(B1 (P )) + kL0 f kCsβ (B1 (P ))
when 0 < β < 1 and α = β/2 and r ≤ 1/2. Let f = R1P f , and apply the interior
remainder estimate to f in Corollary I.8.8 to get
Qr
f kC 0 (Aµr (Qr )) ≤ C kf kC 0 (Aλr (Qr )) + r1+α kL0 f kHsβ (Aλr (Qr )) .
kR2,1
Now any polynomial of degree k is its own Taylor polynomial of degree k at every
point, so
Qr P
T1 f = T1P f
T2,1
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
928
P. DASKALOPOULOS AND R. HAMILTON
and hence
Qr
Qr P
Qr
R2,1
f = R2,1
R1 f = R2,1
f .
Since Aλr (Qr ) ⊆ B2r (P ) we can estimate the C 0 norm of f by the boundary
remainder estimate given before. Since f differs from f by a polynomial of degree 1
in both space and time, L0 f differs from L0 f by only a constant; and this constant
is annihilated by the semi-norm H α . Thus
kL0 f kHsβ (Aλr (Qr )) = kL0 f kHsβ (Aλr (Qr )) .
Combining these estimates gives
Qr
f kC 0 (Aµr (Qr )) ≤ Cr1+α kf kC 0 (B1 (P )) + kL0 f kCsβ (B1 (P )) .
kR2,1
Now note that
Qr
Qr
T2,1
f − T1P f = R1P f − R2,1
f
and
Qr
Qr
kR1P f − R2,1
f kC 0 (Aµr (Qr )) ≤ kR1P f kC 0 (B2r (P )) + kR2,1
f kC 0 (Aµr (Qr ))
so
Qr
kT2,1
f − T1P f kC 0 (Aµr (Qr )) ≤ Cr1+α kf kC 0(B1 (P )) + kL0 f kCsβ (B1 (P )) .
Now it is possible to bound the coefficients of a polynomial from its supremum over
a cylinder of a certain size, and hence to bound the derivatives of the polynomial.
Consider the polynomial
Qr
p = T2,1
f − T1P f
which is bounded on the cylinder Aµr (Qr ), and consider for example its second
derivative fxx with respect to x twice, which is a constant. We have the estimate
C
|pxx | ≤ 2 kpkC 0 (Aµr (Qr )) .
r
But
Qr
T2,1
f
= fxx (Qr )
xx
and
T1P f
xx
=0
so
pxx = fxx (Qr ) .
This gives an estimate on fxx (Qr ), and the same applies to fxy (Qr ) and fyy (Qr ).
This gives the following result.
I.9.1. Theorem. For any ν > 0 and any β in 0 < β < 1 we can find a constant
C with the following property. If f is smooth in the box B1 and if 0 < r ≤ 1/2 and
if
Qr = {x = r, y = 0, t = 1}
and α = β/2, then
|fxx (Qr )| + |fxy (Qr )| + |fyy (Qr )| ≤ Crα−1 kf kC 0 (B1 ) + kL0 f kCsβ (B1 ) .
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
929
By scaling space and time around the point P = {x = 0, y = 0, t = 1} = Q0 and
translating in y and t, we can get a similar estimate at any point in the box B1/4 ,
since we only need to shrink B1 by a factor of at most 1/2, and then r ≤ 1/2 turns
into x ≤ 1/4. When we note that
   
0
x
√
s y  , y  = 2x
t
t
we get the following.
I.9.2. Corollary. We also have
   β−2
0
x
|fxx | + |fxy | + |fyy | ≤ Cs y  , y
kf kC 0(B1 ) + kL0 f ||Csβ (B1 )
t
t
 
x
for all y  in B1/4 .
t
I.9.3. Corollary. We also have
   β
x
0
|xfxx | + |xfxy | + |xfyy | ≤ Cs y  , y  kf kC 0(B1 ) + kL0 f ||Csβ (B1 ) .
t
t
Proof. Because 2x = s2 . This is a Hölder estimate on xfxx , xfxy , xfyy for shifts in
x only, since they are 0 at x = 0.
I.9.4. Corollary. We also have
 
   
   
 
x
0 x
0 x
0 fx y  − fx y  + fy y  − fy y  + ft y  − ft y 
t
t t
t t
t    β
x
0
≤ Cs y  , y  kf kC 0 (B1 ) + kL0 f ||Csβ (B1 ) .
t
t
Proof. We integrate
and use
and
 
 
 
Z x
x
0
u
fx  y  − fx y  =
fxx y  du
u=0
t
t
t
 
u fxx y  ≤ Cuα−1 kf kC 0 (B ) + kL0 f k β
1
Cs (B1 )
t Z
x
xα
α
u=0
to get the bound on fx . The bound on fy follows by integrating fxy the same way.
The bound on ft follows from the equation
uα−1 du =
ft = x(fxx + fyy ) + νfx + g
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
930
P. DASKALOPOULOS AND R. HAMILTON
where g = L0 f , and the Hölder bound on g which is given. These are Hölder
bounds on fx , fy and ft for shifts in x only.
We can also make Hölder estimates in the interior using the bound on f = R1P f .
As in the previous argument on the remainder formula we get
1
kf kHs2+β (Aµr (Qr )) ≤ C
kf kC 0 (Aλr (Qr )) + kL0 f kHsβ (Aλr (Qr ))
r1+α
and Aλr (Qr ) ⊆ B2r (P ) and
kf kC 0 (B2r (P )) ≤ Cr1+α kf kC 0 (B1 (P )) + kL0 f kCsβ (B1 (P ))
and
kL0 f kHsβ (Aλr (Qr )) ≤ CkL0 f kCsβ (B1 (P )) .
We also have
kf kHs2+β = kf kHs2+β
since the Hs2+β semi-norm measures the Hsβ norm of first or second derivatives only,
and first derivatives of the first order polynomial
T1P f = f − f
are constant, and another difference or derivative gives zero. Combining these
proves the following result.
I.9.5. Theorem. Under the hypotheses of Theorem I.9.1, we also have
kf kHs2+β (Aµr (Qr )) ≤ C kf kC 0 (B1 (P )) + kL0 f kCsβ (B1 (P )) .
x
I.9.6. Theorem. For all y in Aµr (Qr ) we have
t
 
   
   
 
x
r x
r x
r fx y  − fx 0 + fy y  − fy 0 + ft y  − ft 0
t
1 t
1 t
1  
   
 
x
r x
r + xfxx y  − rfxx 0 + xfxy y  − rfxy 0
t
1 t
1  
 
x
r + xfyy y  − rfyy 0
t
1    β
x
r
≤ Cs y  , 0 kf kC 0 (B1 ) + kL0 f kCsβ (B1 ) .
t
1
I.9.7. Corollary. For any ν > 0 and any β in 0 < β < 1 we can find ε > 0 and a
constant C < ∞ with the following property. If f is a smooth function on B1 and
if
 
 
x1
x2
P1 =  y1 
and
P2 =  y2 
t1
t2
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
931
are any two points in B1/4 and if
|x1 − x2 | + |y1 − y2 | + |t1 − t2 | ≤ ε(x1 + x2 ),
then
|xfxx (P1 ) − xfxx (P2 )| + |xfxy (P1 ) − xfxy (P2 )| + |xfyy (P1 ) − xfyy (P2 )|
≤ Cs(P1 , P2 )β kf ||C 0 (B1 ) + kL0 f kCsβ (B1 ) .
Proof. Assume t2 ≥ t1 (or else switch). Translate y and t so that y2 = 0 and t2 = 1,
and then dilate x, y, t by a factor
1/(1 − t2 ) ≤ 2 so that the new function is still
r
defined on B1 . Now P2 = 0 for some r ≤ 1/2, and if ε > 0 is small enough
1
the new point P1 will lie in Aµr (Qr ) when P2 = Qr now. We can then apply the
previous result, and the constant will have changed by a bounded factor.
I.9.8. Theorem. For any ν > 0 and any β in 0 < β < 1 we can find δ > 0 and a
constant C < ∞ such that if P1 and P2 are any two points in the box Bδ and if f
is any smooth function on the box B1 , then
|fx (P1 ) − fx (P2 )| + |fy (P1 ) − fy (P2 )| + |ft (P1 ) − ft (P2 )|
+ |xfxx (P1 ) − xfxx (P2 )| + |xfxy (P1 ) − xfxy (P2 )|
+ |xfyy (P1 ) − xfyy (P2 )|
≤ Cs(P1 , P2 )β kf kC 0(B1 ) + kL0 f kCsβ (B1 ) .
Proof. By the previous result we are done with δ ≤ 1/4 unless
|x1 − x2 | + |y1 − y2 | + |t1 − t2 | ≥ ε(x1 + x2 )
for some ε > 0. In this case the metric distance
p
p
p
s(P1 , P2 ) ≈ |x1 − x2 | + |y1 − y2 | + |t1 − t2 |
in the sense of being bounded above and below by a constant times the expression.
Choose
r = Bs(P1 , P2 )2
where B is a large constant so that
|y1 − y2 | + |t1 − t2 | ≤ εr
and where δ is so small that r ≤ 1/2 for all P1 and P2 in Bδ for this B. Let
 
 
 
 
0
0
r
r
P3 = y1  ,
P4 = y2  ,
P5 = y1  ,
P6 = y2  .
t1
t2
t1
t2
Our strategy is to compare a function, say fx or xfxx , at P1 with its value at P3 ,
and that with its value at P5 ; likewise its value at P2 with its value at P4 , and
that with its value at P6 ; and finally the value at P5 with the value at P6 . We can
bound all these distances in terms of r
√
s(P1 , P3 ) + s(P3 , P5 ) + s(P2 , P4 ) + s(P4 , P6 ) + s(P5 , P6 ) ≤ C r
√
and r ≤ Bs(P1 , P2 ). The comparisons of the function between P1 and P3 or P3
and P5 or P2 and P4 or P4 and P6 now follow from Corollary I.9.3, while that
between P5 and P6 follows from Corollary I.9.7. Note the path P3 P1 P5 P6 P4 P2
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
932
P. DASKALOPOULOS AND R. HAMILTON
somewhat resembles the cycloid, which is a geodesic in our metric. This gave rise
to this idea.
Now it is possible to bound the Cs2+β norm of f using the Hs2+β norm and the
0
C norm. For example, suppose we wish
x0 to bound the C norm of fx . Let its
maximum M over Bδ occur at a point y0 where
t0
 
x0 fx  y0  = M .
t0 x
We know for all y in Bδ that
t
 
 
x
x0 fx y  − fx  y0  ≤ C kf kC 0(B ) + kL0 f k β
.
1
Cs (B1 )
t
t0 0
Unless
we have
M ≤ 2C kf ||C 0 (B1 ) + kL0 f kCsβ (B1 )
 
x fx y  ≥ M/2
t on all of Bδ . Integrating in x makes
 
 
x
x0 f y0  − f  y0  ≥ M |x − x0 |
2
t0
t0 and since |x − x0 | ≥ δ/2 for some x we get a contradiction if f is bounded and M
is too big. The other bounds follow in the same way. For example, the bound on
xfxx follows from
xfxx = (xfx − f )x .
If xfxx is too large at one point, then xfx − f has too large a variation.
I.9.9. Corollary. We also have
kf ||Cs2+β (Bδ ) ≤ C kf ||C 0 (B1 ) + kL0 f kCsβ (B1 ) .
I.10. Main Schauder estimate. Combining the results in the previous sections
we can prove now our main Schauder estimate.
I.10.1. Theorem (Schauder Estimate). For any ν > 0, any α in 0 < α < 1 and
any r < 1 there is a constant C so that
kf kCs2+α (Br ) ≤ C kf kCs◦(B1 ) + kL0 f kCsα (B1 )
for all C ∞ smooth functions f on B1 .
Proof. The result follows directly; since for any r < 1 we can cover a neighborhood
of the part of the box Br along the boundary {x = 0} with little boxes that translate
and dilate to Bδ as before. Having the estimate in a neighborhood of the boundary
reduces it to an interior problem, which we can handle in the same manner.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
933
The next Corollary can be shown via standard rescaling.
I.10.2. Corollary. For any ν > 0, any α in 0 < α < 1 and any 0 < r < ρ < 1
there is a constant C so that
kf kCs2+α (Br ) ≤ C kf kCs◦(Bρ ) + kL0 f kCsα (Bρ )
for all C ∞ smooth functions f on B1 .
We will now give the proof of Theorem I.1.3.
Proof of Theorem I.1.3. Assume first that k = 1. Let f be a smooth function in
the box B1 and set L0 f = g. The derivatives ft , fx and fy satisfy the equations
(ft )t − x[ (ft )xx + (ft )yy ] − ν(ft )x = gt ,
(fy )t − x[ (fy )xx + (fy )yy ] − ν(fy )x = gy ,
(fx )t − x[ (fx )xx + (fx )yy ] − (ν + 1)(fx )x = gx + fyy
in the box B1 . For any r < 1, set ρ1 = (1 + 2r)/3 and ρ2 = (2 + r)/3. Then
0 < r < ρ1 < ρ2 < 1. From the estimates in Theorem I.10.1 and Corollary I.10.2
we have
kf kCs2+α(Bρ ) ≤ C kf kCs◦(B1 ) + kgkCsα (B1 )
2
and
||ft ||Cs2+α (Bρ
1)
+ ||fy ||Cs2+α (Bρ
and
||fx ||Cs2+α (Br ) ≤ C
1)
≤C
kDf kCs◦(Bρ2 ) + kDgkCsα(Bρ2 )
kDf kCs◦(Bρ1 ) + kDgkCsα(Bρ1 ) + kfyy kCsα (Bρ1 ) .
Combining the estimates above we conclude
kf kCs1,2+α(Br ) ≤ C kf kCs◦(B1 ) + kgkCs1,α(B1 )
which is the desired estimate for k = 1. The same proof, with a bit more involved
notation, generalizes for all k ≥ 1. The constant C in this case depends on the
integer k.
I.11. Smoothing operators and an Extension Lemma. The proof of Theorem I.1.1 will follow from Theorems I.1.2 and I.1.3 via a regularizing argument
which will involve appropriate smoothing operators with respect to the metric s.
We begin by defining these operators.
Let P = ( xy ) be a point on the half-space x ≥ 0 and Q = ( uv ) any point in the
unit box
B1 = {|u| < 1, |v| ≤ 1 }.
For > 0 we define the point M (P ; Q) as follows. Starting from the point x+2
y
√
we first move by a distance |u| (in the s metric) in the direction parallel to the
if u > 0 or u < 0 respectively. Then,
x-axis and to the right or left of x+2
y
√
starting from the new point we move by a distance |v| (again in the s metric) in
the direction of the y-axis and above or below of the new point if v > 0 or v < 0
respectively. Thus if the point M (P ; Q) has coordinates
ξ
M (P ; Q) =
ζ
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
934
we have
P. DASKALOPOULOS AND R. HAMILTON
√
x + 2
ξ
s
,
= |u|,
y
y
√
ξ
ξ
s
,
= |v|
y
ζ
where
ξ ≥ x + 2
iff u ≥ 0
and
ζ ≥ y iff v ≥ 0.
√
√
Since for dy = 0, ds = dx/( 2x) and for dx = 0, ds = dy/( 2x), we compute
(√
p
√
2ξ − 2(x + 2) = u,
√ √
(I.11.1)
ζ − y = 2ξ v.
Let ϕ be a standard smooth, nonnegative bump function, supported in the box B1 ,
with
Z
ϕ(u, v) dudv = 1
and let h = h ( xy ) be a function defined on the half-space S0 where x ≥ 0. We
define the spatial regularizations h of h by
Z
Z dξ
dζ
h (P ) = ϕ ( uv ) h (M (P ; ( uv ))) dudv = ϕ M−1 P ; ζξ
h ζξ
2ξ
for P = ( xy ) ∈ S0 . We begin with the following useful Lemma.
0
I.11.2. Lemma. For any two points P = ( xy ) and P 0 = xy0 in S0 and any point
Q = ( uv ) in B1 , we have
s[ M (P ; Q), M (P 0 ; Q) ] ≤ C s[ P, P 0 ]
and
√
s[ M (P ; Q), P ] ≤ 4 with the constant C independent of .
0
Proof. Let M (P ; Q) = ζξ and M (P 0 ; Q) = ζξ 0 . We can assume that x0 ≥ x.
If x = x0 , then by (I.11.1) ξ = ξ 0 and ζ − y = ζ 0 − y 0 . Therefore, since on |u| ≤ 1
we have
p
p
√
√
2ξ ≥ 2(x + 2) − ≥ x
√
and for dξ = 0, ds = dζ/ 2ξ, we estimate
h 0 i
0 i
√ h
1
1
s ζξ ; ζξ 0
≤ √ |ζ − ζ 0 | ≤ √ |y − y 0 | = 2 s ( xy ) ; xy0 .
x
2ξ
On the other hand, if y = y 0 , then, by (I.11.1), we have
p
p
p
√
√ p
|ζ − ζ 0 | = v | 2ξ − 2ξ 0 | = v| 2(x + 2) − 2(x0 + 2)|
√
√
√
≤ v | 2x − 2x0 |
p
p
since the function ψ() = 2(x + 2) − 2(x0 + 2) is decreasing in . Thus, since
h 0 i
p
p
|ξ − ξ 0 | + |ζ − ζ 0 |
|ζ − ζ 0 |
≤C √
≤ C | ξ − ξ0 | + √ 0
s ζξ ; ζξ 0
√ 0 p
ξ
ξ + ξ + |ζ − ζ 0 |
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
935
we estimate
√ h
0 i
h 0 i
√
√
v
ξ
ξ
0
≤ C | 2x − 2x | 1 + √ 0 ≤ C s ( xy ) ; xy0
s ζ ; ζ0
2ξ
√ 0 √
since ξ ≥ . Since
h
0 i
0 i
h 0 0 i
h
0 ≤ s ( xy ) , xy + s xy , xy0
≤ c s ( xy ) , xy0
s ( xy ) , xy0
for some c > 0, the first estimate follows. As for the second estimate, we compute
] + s[ x+2
;P ]
s[ M (P ; Q), P ] ≤ s[ M (P ; Q), x+2
y
y
p
√
√
√
≤ 2 + 2(x + 2) − 2x ≤ 4 p
√
√
] = 2 and ψ(x) = 2(x + 2) − 2x is
because by definition s[ M (P ; Q), x+2
y
decreasing in x.
We can now give our regularization result in the metric s.
I.11.3. Theorem. If h ∈ Csα (S0 ), then h is smooth on S0 ,
||h ||Csα (S0 ) ≤ C ||h||Csα (S0 )
and for all points
( xy )
in S0
|h ( xy ) − h ( xy ) | ≤ Cα/2 ||h||Csα (S0 ) .
Therefore h → h, uniformly on S0 .
Proof. We first notice that the point M (P ; Q) = ζξ , with P ∈ S0 and Q ∈ B1 has
√
√
2ξ ≥ . Hence, each function h is smooth on S0 . It is clear that ||h ||C0 ≤ ||h||C0 .
Also, by the previous lemma, for P, P 0 ∈ S0 we have
Z
|h (P ) − h (P 0 )| ≤ ϕ(Q)|h(M (P ; Q)) − h(M (P 0 ; Q))| ≤ C ||h||Csα (S0 ) s[ P ; P 0 ]α
R
since ϕ = 1. Thus, ||h ||Csα (S0 ) ≤ C ||h||Csα (S0 ) , for some constant C independent
of . Moreover, for any P ∈ S0 we have
Z
|h (P ) − h(P )| ≤ ϕ(Q)|h(M (P ; Q)) − h(P )| ≤ ||h||Csα (S0 ) s[ M (P ; Q), P ]α
√
with s[ M (P ; Q), P ] ≤ 4 , as we proved in Lemma I.11.2. Therefore h → h,
uniformly on S0 .
We continue with an Extension Lemma on the new Hölder spaces. Such a result is
standard for regular Hölder spaces. We denote, as usual by S the space S0 × [0, ∞).
I.11.4. Theorem. Assume that g ∈ Csα (S) and f 0 ∈ Cs2+α (S0 ), for some number
α in 0 < α < 1. Then, there exists a function h ∈ Cs2+α (S) such that
x
x
∂h x y
=g y
and
h y = f 0 ( xy )
0
0
∂t 0
and
||h||Cs2+α (S) ≤ C ||f 0 ||Cs2+α (S0 ) + ||g||Csα (S)
for some constant C depending only on α.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
936
P. DASKALOPOULOS AND R. HAMILTON
Proof. The result can be easily reduced to the case that f 0 ≡ 0 in S0 . We will show
that in this case the function
Z
x Z t
h y =
dτ ϕ ( uv ) g (Mτ (( xy ) ; ( uv ))) du dv
t
0
defined on S has the desired properties. It is easy to show that
x
x
∂h x y
and
=g y
h y =0
0
0
∂t 0
since g is a continuous function. Hence, it remains to show that h belongs to the
space Cs2+α (S) and satisfies
||h||Cs2+α (S) ≤ C ||g||Csα (S) .
The proof of such an estimate is standard for Hölder spaces in the Euclidean metric.
The calculations in our Hölder spaces are a bit more involved but similar. We will
only show that xhxx belongs in the space Csα (S) and satisfies the estimate
||xhxx ||Csα (S) ≤ C||g||Csα (S) .
All the other estimates can be derived in a similar manner.
We first estimate the L∞ norm of xhxx . Changing variables we can write h as
x Z t Z dξ
dζ.
h y =
g ζξ
dτ ϕ Mτ−1 ( xy ) ; ζξ
t
2τ ξ
0
If we set
Mτ−1 ( xy ) ; ζξ
= ( uv ) ,
then by (I.11.1) we have
p
√
2ξ − 2(x + 2τ )
ζ −y
√
and
v= √ √
u=
τ
2ξ τ
and therefore find that
Z t Z x
dξ
x
ϕuu + √
dζ.
xhxx =
dτ
ϕu g ζξ
3/2
2τ
(x
+
2τ
)
2τ
ξ
2 2τ (x + 2τ )
0
On the other hand, since the function ϕ is compactly supported in B1 , integration
by parts implies that
Z
Z
dξ
dξ
−1
(I.11.5)
dζ = ϕuu (Mτ−1 )
dζ = 0.
ϕu (Mτ )
2τ ξ
2τ ξ
Hence we can express xhxx as
xhxx = I1 + I2 ,
with
Z
I1 =
and
Z
I2 =
dτ
0
Z
t
dτ
0
Z
t
h x
ϕuu (Mτ−1 ) g ζξ − g
2τ (x + 2τ )
x+2τ
y
h x
√
ϕu (Mτ−1 ) g ζξ − g
2 2τ (x + 2τ )3/2
But on the support of ϕu (Mτ−1 (P )) we have
| g ζξ − g x+2τ
| ≤ ||g||Csα s[ ζξ ,
(I.11.6)
y
x+2τ
y
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
i dξ
dζ
2τ ξ
x+2τ
y
i dξ
dζ.
2τ ξ
]α ≤ C||g||Csα τ α/2 .
REGULARITY OF THE FREE BOUNDARY
937
Thus, since x/(x + 2τ ) ≤ 1, we can estimate
Z t α/2 Z
dξ
τ
dτ (|ϕu | + |ϕuu |)(Mτ−1 )
dζ ≤ C tα/2 ||g||Csα (S) .
|xhxx | ≤ C||g||Csα
τ
2τ ξ
0
0
x
x
We continue by estimating the Csα seminorm of xhxx . Let P = y and P 0 = y0
t
t0
be any two points in S. We will present the computation in the case that y = y 0
and t = t0 . All the other cases are similar. We write
xhxx (P ) − x0 hxx (P 0 ) = I + J,
with
Z
I=
0
and
t
dτ
2τ
Z x0
x
dξ
ϕuu (Pe ) − 0
ϕuu (Pe 0 ) g ζξ
dζ
x + 2τ
x + 2τ
2τ ξ
Z x0
dξ
x
0
e
e
dζ
J=
ϕ (P ) − 0
ϕ (P ) g ζξ
3/2 u
3/2 u
2τ
ξ
(x
+
2τ
)
(x
+
2τ
)
0
where Pe = Mτ−1 (P ) and Pe 0 = Mτ−1 (P 0 ). We will estimate I, since J can be
estimated in a very similar manner. Set η = s[ P, P 0 ]2 /4 and split I into the
following four terms:
Z Z t
h i dξ
x0
dτ
x
I1 =
− 0
ϕuu (Pe) g ζξ − g x+2τ
dζ,
y
x + 2τ
x + 2τ
2τ ξ
η 2τ
Z
Z t
ih i dξ
dτ
x0 h
ξ
0
x+2τ
e
e
I2 =
ϕ
dζ,
−
g
(
P
)
−
ϕ
(
P
)
g
uu
uu
y
ζ
x0 + 2τ
2τ ξ
η 2τ
Z
Z η
h i dξ
dτ
x
ϕuu (Pe) g ζξ − g x+2τ
dζ
I3 =
y
x + 2τ
2τ ξ
0 2τ
and
Z
Z η
h i dξ
dτ
x0
e0 ) g ξ − g x0 +2τ
I4 =
ϕ
dζ.
(
P
uu
y
ζ
x0 + 2τ
2τ ξ
0 2τ
Here we made use of identities (I.11.5). We claim that
Z
t
dτ
√
2 2τ
Ii ≤ C||g||Csα s[ P, P 0 ]α ,
i = 1, . . . , 4.
We begin by estimating I1 . Since
Z x
x0
x
2τ
− 0
=
dx̄,
x + 2τ
x + 2τ
(x̄
+
2τ )2
x0
and (I.11.6) holds on the support of ϕuu (Pe ), while
Z
dξ
dζ ≤ C,
|ϕuu |(Mτ−1 (P ))
2τ ξ
we have
Z x Z t
τ α/2
α
dx̄
dτ .
|I1 | ≤ C||g||Cs 2
(x̄
+
2τ
)
0
x
η
Substituting in the last integral τ = x̄s, we easily conclude that
Z x
√
√
−1+α/2
x̄
dx̄ ≤ C ||g||Csα | x − x0 |α
|I1 | ≤ C ||g||Csα 0
x
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
938
P. DASKALOPOULOS AND R. HAMILTON
√
√
from which the desired estimate follows since s[ P, P 0 ] = | 2x − 2x0 |. The
estimate of I2 is similar. We write this time
Z x
1
ϕuu (Pe) − ϕuu (Pe 0 ) =
ϕuuu (Mτ−1 (P )) p
dx̄,
2τ (x̄ + 2τ )
x0
x
with P = y , and instead of (I.11.6) use the estimate
t
| g ζξ − g x+2τ
| ≤ ||g||Csα τ α/2 + s[ P, P 0 ]α ≤ C ||g||Csα τ α/2
y
holding on the union of the supports of ϕuu (Pe) and ϕuu (Pe0 ) and for τ ≥ η =
s[P, P 0 ]2 /4. Since
Z
dξ
dζ ≤ C
|ϕuuu |(Mτ−1 (P ))
2τ ξ
we have
Z
|I2 | ≤ C||g||Csα Z
x
dx̄
x0
η
t
x̄τ (α−1)/2
dτ
3/2
(x̄ + 2τ )
which, under the substitution τ = x̄s in the second integral, gives us the desired
estimate. It remains to estimate I3 , since the estimate for I4 follows by symmetry.
Because once more (I.11.6) holds on the support of ϕu (Pe) we conclude
Z η α/2
τ
dτ ≤ C ||g||Csα η α/2
|I3 | ≤ C ||g||Csα
τ
0
which immediately implies the desired estimate, since η = s[ P, P 0 ]2 /4. The rest of
the calculations are similar.
We can extend the previous result to Hölder spaces of higher order derivatives.
I.11.7. Theorem. Assume for some nonnegative integer k and some number α in
0 < α < 1, that g ∈ Csk,α (S) and f 0 ∈ Csk,2+α (S0 ). Then, there exists a function
h ∈ Csk,2+α (S) such that
x
x
∂h x y
=g y
and
h y = f 0 ( xy )
0
0
∂t 0
and
||h||Csk,2+α (S) ≤ C ||f 0 ||Csk,2+α (S0 ) + ||g||Csk,α (S)
for some constant C depending only on α and k.
Proof. As before, the result can be easily reduced to the case that f 0 ≡ 0. It can
be shown now, via similar computations as in the proof of Theorem I.11.4, that if
g ∈ Csk,α (S), the function h defined, as before, by
Z
x Z t
h y =
dτ ϕ ( uv ) g (Mτ (( xy ) ; ( uv ))) du dv
t
belongs in the space
0
Csk,2+α (S)
and satisfies
||h||Csk,2+α (S) ≤ C ||g||Csk,α (S) .
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
939
Before we finish
in space and
x this section we will introduce smoothing operators
u
e
e
time. Let P = y be a point in S = S0 × [0, ∞) and Q = v any point in the
s
t
f1 = {|u| < 1, |v| ≤ 1, |s| ≤ 1 }. For > 0 let M denote the spatial
unit box B
regularizations introduced in the beginning of this section. Starting from the point
Pe, we define now the new point
 
ξ
e = ζ  ,
f (Pe; Q)
M
τ
having
ξ
= M (( xy ) ; ( uv )) ,
ζ
and
τ = t + 2 + s.
The following Lemma is a direct consequence of the above definitions and Lemma
I.11.2.
e in the box B
e1 ,
I.11.8. Lemma. For any two points Pe , Pe 0 in S and any point Q
we have
f (Pe; Q),
e M
f (Pe0 ; Q)
e ] ≤ C s[ Pe , Pe 0 ]
s[ M
and
f (Pe ; Q),
e Pe ] ≤ C
s[ M
√
.
Now let g be a continuous function on S and let ϕ
e be a standard smooth,
f
nonnegative bumb function, supported in B1 , and such that
Z
Z u
ds ϕ
e v dudv = 1.
s
We define the regularizations g of g as
g (Pe) =
Z
Z
ds
e g(M
f (Pe ; Q))
e dudv =
ϕ(
e Q)
Z
dτ
Z
f−1 (Pe; R))
e g(R)
e
ϕ(M
dξ
dζ
2ξ
e = M (Pe ; Q).
e As an immediate consequence of Theorem I.11.3, we obtain
where R
the following space-time regularizing result:
I.11.9. Theorem. For any function g in Csα (S) and any two points Pe and Pe 0 in
S, we have
||g ||Csα (S) ≤ C ||g||Csα (S)
and
|g (Pe) − g(Pe 0 )| ≤ Cα/2 ||g||Csα (S)
with C independent of .
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
940
P. DASKALOPOULOS AND R. HAMILTON
I.12. Existence and uniqueness. In this last section of Part I we will give the
proof of the existence and uniqueness Theorem I.1.1.
Proof of Theorem I.1.1. We begin with the existence question. Because of Theorem
I.11.7, we can assume without loss of generality that f 0 ≡ 0 and that g is a function
in Csk,α (S) such that
x
g y =0
∀ ( xy ) ∈ S0 .
0
Let g be the space-time regularizations of the function g, as defined at the end of
the previous section. Each g is smooth, compactly supported in S = S0 × [0, ∞)
and vanishes at t = 0. In addition, it follows from Theorem I.11.9, that
||g ||Csk,α (S) ≤ C ||g||Csk,α (S)
and
g → g,
as → 0
uniformly on S. Let f be the unique solution of the initial value problem
(
in S,
L 0 f = g
f (·, 0) = 0
on S0
satisfying
||f ||C 0 (S) ≤ C ||g ||C 0 (S)
as constructed in Theorem I.1.2. The Schauder estimate in Theorem I.1.3 implies
that for every compact subset K of S we have
||f ||Csk,2+α (K) ≤ C ||g ||Csk,α (S)
with C independent of K. Let 0 < β < α. It follows from the previous estimate
that there exists a subsequence {fn } of {f } which converges in C k,2+β (K) to a
function f , for all compact subsets K of S. It is now easy to conclude that f belongs
to the space C k,2+α (S) and satisfies
||f ||C k,2+α (S) ≤ C ||g||Csk,α (S)
as desired.
The uniqueness of solutions follows from the classical maximum principle, as in
Theorem I.3.1. After a moment of thought we come to the conclusion that the
maximum of a solution f of the equation
ft − x ( fxx + fyy ) − νfx = 0
cannot occur at the boundary x = 0 since
x fxx = x fyy = 0,
at x = 0
Csk,2+α (S)
and ν > 0. The last identities obviously hold
for all functions f ∈
true when k ≥ 1, in which case every function f in Csk,2+α (S) is at least twice
differentiable up to the boundary x = 0. When f ∈ Cs2+α (S) we still have x fxx =
x fyy = 0 at x = 0, as shown next.
I.12.1. Proposition. Every function f ∈ Cs2+α (S) satisfies
xfxx = xfyy = 0,
at x = 0.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
941
Proof. We will show that xfxx = 0 at x = 0, as the rest of the identities follow via
similar arguments. We will proceed by contradiction. Assume that for some point
−∞ < y0 < ∞ we have
lim x fxx (x, y0 ) = a 6= 0
x→0
and assume that a > 0, otherwise replace f by −f . Since the function x fxx is
continuous up to the boundary x = 0, there exists a number δ > 0 such that
a
x fxx (x, y) ≥
2
for all 0 ≤ x ≤ δ, y0 − δ ≤ y ≤ y0 + δ. But then
Z δ
Z δ
a
dx = ∞
fxx (x, y0 ) dx ≥
fx (δ, y0 ) − fx (0, y0 ) =
2x
0
0
is impossible, since fx is continuous up to the boundary x = 0. Therefore, xfxx = 0
at x = 0, finishing the proof of the proposition.
We conclude this section with the following generalization of Theorem I.1.1 which
will be used in Part II of the paper. For a number T > 0 we denote, as before, by
ST the space S0 × [0, T ].
I.12.2. Theorem. Let k be a nonnegative integer and α a number in 0 < α < 1.
Assume that g ∈ Csk,α (S) and f 0 ∈ Csk,2+α (S0 ), with both g and f 0 compactly
supported in S and S0 respectively. Then, for any constant c and any ν > 0 and
T > 0, the initial value problem
(
L0 f − cf = g
in ST ,
f (·, 0) = f 0
on S0
admits a unique solution f ∈ Csk,2+α (ST ) which satisfies the estimate
||f ||Csk,2+α (ST ) ≤ C(T ) ||f 0 ||Csk,2+α (S0 ) + ||g||Csk,α (ST )
for some constant C(T ) depending on k, α, ν, c and T .
Proof. The uniqueness assertion of the Theorem follows, as in Theorem I.3.1, from
a direct consequence of the classical maximum principle.
For existence, you just need to observe that f is the desired solution if and only
if f˜ = e−ct f is a solution to the problem
(
in ST ,
L0 f˜ = e−ct g
f (·, 0) = f 0
on S0
which can be solved by Theorem I.1.1. It is clear that the constant C(T ) will also
depend on c.
Part II. Degenerate equations with variable coefficients
II.1. The linear case. In this section we will study linear degenerate equations
of the form
wt = (ϑaij wij + bi wi + c w) + g
on the cylinder Ω × [0, ∞), where Ω is a compact domain in R2 with smooth
boundary. The subindices i, j ∈ {x, y} denote differentiation with respect to the
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
942
P. DASKALOPOULOS AND R. HAMILTON
space variables x, y and the summation convention is used. We assume that the
coefficient matrix (aij ) is strictly positive and all coefficients aij , bi and c belong
to appropriate Hölder spaces which will be defined later.
The degeneracy of the equation is carried through the function ϑ(x) which is
assumed to be smooth on Ω, strictly positive in its interior, with
ϑ(P ) = dist (P, ∂Ω)
for all P ∈ Ω sufficiently close to ∂Ω. We also assume that
bi ni > 0 on ∂Ω
where n = (n1 , n2 ) denotes the interior normal to the boundary of Ω and again the
summation convention is used.
When the boundary is flat and the coefficients are constants, this equation (locally and under an appropriate change of variables) takes the form of the model
equation studied in Part I
ft = x (fxx + fyy ) + ν fx + g
on the half-space x ≥ 0. The condition bi ni > 0 at ∂Ω is then equivalent to the
condition ν > 0. Notice that, as in the constant coefficient model, we don’t impose
any conditions on w along the boundary of Ω, where the principal coefficients ϑaij
vanish. Because of the condition bi ni > 0 at ∂Ω, the lateral boundary of the
cylinder Ω × [0, ∞) behaves like a free boundary.
Imitating the model case where the operators are defined on the half-space {x ≥
0}, we define the cycloidal distance function s in Ω. In the interior of Ω the cycloidal
distance will be equivalent to the standard Euclidean distance, while around any
point P ∈ ∂Ω, s is defined as the pull back of the cycloidal distance on the halfspace S = {x ≥ 0}, as defined in Part I, via a map ϕ : S → Ω that straightens the
boundary of Ω near P .
It can be easily shown that the cycloidal distance between two points P1 = ( xy11 )
and P2 = ( xy22 ) in Ω is equivalent to the function
|P1 − P2 |
p
p
s̄(P1 , P2 ) = p
|P1 − P2 | + d(P1 ) + d(P2 )
with d = d(P ) denoting the distance to the boundary of Ω.
The parabolic distance in the cycloidal metric is equivalent to the function
p
s̄ Pt11 , Pt22 = s(P1 , P2 ) + |t1 − t2 |.
Now suppose that A is a subset of the cylinder Ω × [0, ∞) which is the closure
of its interior. As in Part I, we denote by Csα (A) the space of Hölder continuous
functions on A with respect to the metric s and by Cs2+α (A) the space of all functions
w on A such that w, wt , wi and d wij , with i, j ∈ {x, y} and with d denoting the
distance function to the boundary of Ω, extend continuously up to the boundary
of A and the extensions are Hölder continuous on A of class Csα (A). They are both
Banach spaces under the norms ||w||Csα (A) , as defined in Part I, and
||w||Cs2+α (A) = ||w||Csα (A) + ||wi ||Csα (A) + ||d wij ||Csα (A)
respectively, where once more the summation convention is used. Also, in analogy
to the corresponding definitions in Part I, we denote by Csk,α (A) and Csk,2+α (A) the
spaces of all functions w whose k-th order derivatives Dxi Dyj Dtl w with i + j + l = k
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
943
exist and belong to the spaces Csα (A) and Cs2+α (A) respectively. Both spaces
equipped with the norms
X
||w||Csk,α (A) =
||Dxi Dyj Dtl w||Csα (A)
i+j+l≤k
and
||w||Csk,2+α (A) =
X
i+j+l≤k
||Dxi Dyj Dtl w||Cs2+α (A)
respectively, are Banach spaces. We will denote by Csα (A) and Cs2+α (A) the spaces
Cs0,α (A) and Cs0,2+α (A) respectively.
Denoting by L the operator
Lw = wt − (ϑaij wij + bi wi + c w),
by Ωσ , for σ > 0, the set
Ωσ = { x ∈ Ω : dist (x, ∂Ω) ≥ σ },
by QT , for T > 0, the cylinder Ω ×[0, T ] and, as always, by n = (n1 , n2 ) the interior
unit normal to ∂Ω, we can now state the main result in Section II.1:
II.1.1. Theorem (Existence and Uniqueness). Let Ω be a compact domain in R2
with smooth boundary and let k be a nonnegative integer, a a number in 0 < a < 1
and T a positive number. Assume that the coefficients aij , bi and c of the operator
L belong to the space Csk,α (QT ) and satisfy the ellipticity condition
aij ξi ξj ≥ λ|ξ|2 > 0
∀ξ ∈ R2 \ {0}
and the bounds
||aij ||Csk,α (QT )
||bi ||Csk,α (QT )
||c||Csk,α (QT ) ≤ 1/λ
and
b i ni ≥ ν > 0
on
∂Ω × [0, T ]
for some positive constants λ and ν. In addition, assume that ϑ is a smooth function
on Ω, strictly positive in its interior, with ||ϑ||C ∞ (Ω) ≤ 1 and such that
ϑ(P ) = dist (P, ∂Ω)
∀P ∈ Ω \ Ωσ
for some σ > 0. Then, given any funtion w0 ∈ Csk,2+α (Ω) and any function
g ∈ Csk,α (QT ) there exists a unique solution w ∈ Csk,2+α (QT ) of the initial value
problem
(
Lw = g
in QT ,
w(·, 0) = w0 on Ω
satisfying
||w||Csk,2+α (QT ) ≤ C(T ) ||w0 ||Csk,2+α (Ω) + ||g||Csk,α (QT ) .
The constant C(T ) depends only on the domain Ω and the numbers α, k, λ, ν, σ
and T .
Proof. Because of the extension Theorem I.11.7 we can assume, without loss of
generality, that f ≡ 0 and that g is a function in Csk,α (QT ), which vanishes at
t = 0.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
944
P. DASKALOPOULOS AND R. HAMILTON
k,2+α
k,α
For δ > 0, set Qδ = Ω × [0, δ] and denote by Cs,0
(Qδ ) and Cs,0
(Qδ ) the
k,2+α
k,α
(Qδ ) and Cs (Qδ ) respectively, consisting of all functions which
subspaces of Cs
k,α
(Qδ ). We
vanish identically at t = 0. Also, denote by I the identity operator on Cs,0
k,α
will show that, if δ is sufficiently small, there exists an operator M : Cs,0 (Qδ ) →
k,2+α
Cs,0
(Qδ ) such that
|| L M − I || ≤
1
.
2
k,α
k,α
This will immediately imply that the operator L M : Cs,0
(Qδ ) → Cs,0
(Qδ ) is
k,2+α
k,α
invertible and therefore L : Cs,0
(Qδ ) → Cs,0
(Qδ ) will be onto, as desired.
We begin by expressing the compact domain Ω as the finite union


[
Ω = Ω0 ∪  Ωl 
l≥1
of compact domains in such a way that
dist (Ω0 , ∂Ω) ≥
ρ
>0
2
and for all l ≥ 1
Ωl = Bρ (xl ) ∩ Ω
with Bρ (xl ) denoting the ball centered at xl ∈ ∂Ω of radius ρ > 0. The number
ρ > 0 will be determined later.
The operator L, when restricted on the interior domain Ω0 is nondegenerate.
Therefore, the classical Schauder theory for linear parabolic equations implies that
L is invertible when restricted on functions which vanish outside Ω0 . Notice that
our Hölder spaces with respect to the cycloidal metric s on the interior domain
Ω0 coincide with the standard Hölder spaces, where the classical Schauder theory
k,α
k,2+α
(Ω0 × [0, δ]) → Cs,0
(Ω0 × [0, δ]) the inverse of
holds true. Denote by M0 : Cs,0
the operator L restricted on Ω0 .
Next, we concentrate our attention on the domains Ωl , l ≥ 1, close to the boundary of Ω, which can be chosen in such a way that the sets Bρ/4 (xl ) ∩ Ω are disjoint.
Denoting by B the half unit ball
B = {(x, y) ∈ B1 (0) ; x ≥ 0 }
and by Qδ the cylinder
Qδ = B × [0, δ]
we select smooth charts Υl : B → Ωl , which flatten the boundary of Ω, i.e., they
map B ∩{x = 0} onto Ωl ∩∂Ω and have Υl (0) = xl . This is possible if the number ρ
is chosen sufficiently small. Under the change of coordinates induced by the charts
Υl , the operator L, restricted on each Ωl × [0, δ], is transformed to an operator L̄l
of the form
i
L̄l w̄ = w̄t − ( x āij
l w̄ij + b̄l w̄i + c̄l w̄)
defined on B × [0, δ]. Moreover, the charts Υl can be chosen appropriately so that
the coefficients of L̄l satisfy
2
āij
l ξi ξj ≥ λ|ξ| > 0
∀ξ ∈ R2 \ {0}
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
945
and
||āij
l ||Csk,α (Q̄δ ) ,
||b̄il ||Csk,α (Q̄δ ) ,
||c̄l ||Csk,α (Q̄δ ) ≤ 1/ λ̄
and
b̄il ≥ ν̄ > 0,
at x = 0
for some positive constants λ̄ and ν̄, while at the point (0, 0) we have
āij
l (0, 0) = 0,
22
ā11
l (0, 0) = āl (0, 0),
for i 6= j
and
b̄1l (0, 0) ≥ ν̄ > 0,
b̄2l (0, 0) = 0.
Here the index i = 1 corresponds to the variable x and j = 2 to the variable y. The
continuity of the coefficients then implies that the constant coefficient operator
e l w̄ = w̄t − [ x āl ( w̄xx + w̄yy ) + ν̄l w̄x + c̄l w̄ ]
L
having
22
āl = a11
l (0, 0) = al (0, 0),
ν̄l = b̄1l (0, 0),
c̄l = c̄l (0, 0)
when defined on Qδ = B × [0, δ] has coefficients sufficiently close to the coefficients
of L̄l in the space Csk,α (Qδ ), if ρ and δ are chosen sufficiently small, depending only
on Ω and the constants σ and λ in the statement of the theorem. Notice that each
e l has the form of the model operators studied in Part I.
of the operators L
Denote, as in Part I, by S0 the half-space x ≥ 0 in R2 and by Sδ the space
k,α
k,α
S0 × [0, δ]. Also, consider the subspace C s,0 (Sδ ) of Cs,0
(Sδ ), consisting of functions
which are compactly supported on Sδ . Then, Theorem I.12.2 implies that for every
fl : C k,α (Sδ ) → C k,2+α (Sδ ) such that
l = 1, 2, ... there is an operator M
s,0
s,0
el M
fl = I
L
k,α
with I denoting the identity operator on C s,0 (Sδ ). Denote by Ml the pull back of
fl via the chart Υl . Next, choose a nonnegative partition of unity
the operator M
φl , l = 0, 1, ... , subordinated to the cover Ωl , l = 0, 1, ... , of Ω and also choose, for
each l ≥ 0, nonnegative, smooth bump functions
ψl , 0 ≤ ψl ≤ 1, supported in Ωl
P
with ψl ≡ 1 on the support of φl . Then l≥0 φl = 1 and ψl φl = φl for all l.
k,α
k,2+α
(Qδ ) → Cs,0
(Qδ ) defined as
Our goal is to show that the operator M : Cs,0
X
Mg =
ψl Ml φl g
satisfies
|| LM g − g ||Csα (Qδ ) <
1
||g||Csα (Qδ )
2
k,α
∀g ∈ Cs,0
(Qδ )
if the cover {Ωl } and δ are chosen appropriately. Indeed, we can write
L Mg − g =
X
l
L ψl Ml φl g −
X
l
φl g =
X
ψl (LMl − I) φl g +
l
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
X
l
[ L, ψl ] Ml φl g
946
P. DASKALOPOULOS AND R. HAMILTON
with [ L, ψl ] denoting the commutator of L and ψl . The commutator [ L, ψl ] is only
of first order and it can be estimated as
||[ L, ψl ] Ml φl g||Csk,α (Qδ ) ≤ C ||ϑD(Ml φl g)||Csk,α (Qδ ) + ||Ml φl g||Csk,α (Qδ ) .
Let > 0. Since the function ϑ is proportional to the distance d to the boundary
of Ω, it follows by the standard interpolation between Hölder spaces that
||ϑD(Ml φl g)||Csk,α (Qδ ) ≤ ||Ml φl g||Csk,2+α (Qδ ) + C() ||Ml φl g||C k,0 (Qδ ) .
However, for each k we have
||Ml φl g||Csk,2+α (Qδ ) ≤ C ||g||Csk,α (Qδ )
and therefore, since Ml φl g ≡ 0 at t = 0,
||Ml φl g||Csk,0 (Qδ ) ≤ C δ ||g||Csk,α (Qδ ) .
Therefore if we choose δ sufficiently small we can make
X
1
|| [ L, ψl ] Ml φl g ||Csk,α (Qδ ) ≤ ||g||Csk,α (Qδ ) .
4
l
On the other hand we have ( LM0 − I )ϕ0 g = 0, while for l ≥ 1, we can make the
norm of each of the operators LMl − I arbitrarily close to zero by choosing the
diameters of the domains Ωl sufficiently small. Wherein
X
1
ψl (L Ml − I) φl g ||Csk,α (Qδ ) < ||g||Csk,α (Qδ )
||
4
l
Csk,α (Qδ ),
for all g ∈
if ρ and δ are both sufficiently small. Combining the above
estimates we obtain that
1
|| L M g − g ||Csk,α (Qδ ) ≤ ||g||Csk,α (Qδ )
2
k,α
k,α
for all g ∈ Cs,0
(Qδ ), as desired. We conclude that for every g ∈ Cs,0
(Qδ ) there
k,2+α
exists a function w ∈ Cs,0
(Qδ ) such that Lw = g. In addition
||w||Csk,2+α (Qδ ) ≤ C ||g||C k,α (Qδ )
with C depending only on Ω and the constants ν, λ and σ. This proves short time
existence. The long time existence follows immediately by the last estimate.
The uniqueness of solutions follows by applying the classical maximum principle
as in Theorem I.3.1.
We give next the generalization of the local Schauder estimates in Theorem I.1.3
for variable coefficient equations. For simplicity we will assume that the operator
L has the form
Lw = wt − (xaij wij + bi wi + cw)
defined on the half-space x ≥0. As
at the beginning of Part I, we define the box
x0 of side r around a point P = y0 to be
t0
 

x ≥ 0, |x − x0 | ≤ r
 x
|y − y0 | ≤ r
Br (P ) = y  :


t 0 − r ≤ t ≤ t0
t
0
and let Br be the box around the point P = 0 . We have the following Theorem:
1
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
947
II.1.2. Theorem. Assume that the coefficients aij , bi and c of the operator L
belong to the space Csα (B1 ), for some number α in 0 < α < 1 and satisfy
aij ξi ξj ≥ λ|ξ|2 > 0
∀ξ ∈ R2 \ {0}
and
||aij ||Csα (QT ) ,
||bi ||Csα (QT ) ,
||c||Csα (QT ) ≤ 1/λ
and
b1 ≥ ν > 0
at x = 0
for some positive constants λ and ν. Then, there exists a constant C depending
only on α, λ, ν such that
kf kCs2+α(B1/2 ) ≤ C kf kCs◦(B1 ) + kLf kCsα(B1 )
for all functions f ∈ Cs2+α (B1 ).
Proof. We will assume that f is a C ∞ function on B1 . The case f ∈ Cs2+α (B1 ) will
then follow via a standard approximation argument, using the smoothing operators
introduced in Section I.11.
Choose a bump function ϕ so that
(
ϕ = 1 on
B1/2 ,
ϕ = 0 off
B1 .
Then, the smooth function ϕf , which vanishes off B1 , satisfies the equation
L(ϕf ) = ϕLf + [L, ϕ]f
with [L, ϕ] denoting the commutator of L and ϕ. Therefore, according to Theorem
II.1.1, ϕf satisfies the estimate
||ϕf ||Cs2+α (B1 ) ≤ C ||ϕ Lf ||Csα (B1 ) + ||[L, ϕ]f ||Csα (B1 ) .
The commutator [L, ϕ] is only of first order and can be estimated, as in the proof
of Theorem II.1.1, by standard interpolation between Hölder spaces. Indeed, for
any number > 0 we have
||[L, ϕ]f ||Csα (B1 ) ≤ ||f ||Cs2+α (B1 ) + C()||f ||C 0 (B1 ) .
Therefore, choosing sufficiently small and remembering that ϕ ≡ 1 on B1/2 , we
obtain
||f ||Cs2+α (B1/2 ) ≤ C ||Lf ||Csα (B1 ) + ||f ||C ◦ (B1 )
as desired.
The next result follows from the Schauder estimate above via a standard rescaling
argument.
II.1.3. Theorem. Under the same hypotheses as in Theorem II.1.2 and for any
number r ≤ 1 there exists a constant C(r) so that
kf kCs2+α(Br/2 ) ≤ C(r) kf kCs◦(Br ) + kLf kCsα(Br ) .
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
948
P. DASKALOPOULOS AND R. HAMILTON
II.2. The quasilinear case. In this section we will the study quasilinear degenerate equations of the form
wt = ϑ F ij (t, x, y, w, Dw) wij + G(t, x, y, w, Dw)
on the cylinder QT = Ω × [0, T ], T > 0. Let us denote by P the operator
P w = ϑ F ij (t, x, y, w, Dw) wij + G(t, x, y, w, Dw)
and by M the operator
M w = wt − P w.
Then, if w̄ is a fixed point in Cs2+α (QT ), the linearization of the operator M at the
point w̄ is the operator
f(w̃) = DM (w̄)(w̃) = w̃t − DP (w̄)(w̃)
M
with
DP (w̄)(w̃) = ϑ F ij (t, x, y, w̄, Dw̄) w̃ij
+ [ ϑ Fwijl (t, x, y, w̄, Dw̄) w̄ij + Gwl (t, x, y, w̄, Dw̄) ] w̃l
+ [ ϑ Fwij (t, x, y, w̄, Dw̄) w̄ij + Gw (t, x, y, w̄, Dw̄) ] w̃.
Here Dw = (w1 , w2 ), Fwij = ∂F ij /∂w, Gw = ∂G/∂w and for l = 1, 2, Fwijl =
∂F ij /∂wl , Gwl = ∂G/∂wl . As always, the summation convention is used. We have
the following Theorem:
II.2.1. Theorem. Assume that Ω is a compact domain in R2 with smooth boundary, let k be a nonnegative integer, and let 0 < α < 1, T > 0 be positive numbers.
Also, let w0 be a function in Csk,2+α (Ω). Assume that the linearization DM (w̄) of
the quasilinear operator
M w = wt − ϑ F ij (t, x, y, w, Dw) wij − G(t, x, y, w, Dw)
defined on QT = Ω × [0, T ], satisfies the hypotheses of Theorem II.1.1 at all points
w̄ ∈ Csk,2+α (QT ), such that ||w̄ − w0 ||Csk,2+α (QT ) ≤ µ, µ > 0. Then, there exists a
number τ0 in 0 < τ0 ≤ T depending on the constants α, k, λ, ν and µ, for which
the initial value problem
(
in Ω × [0, τ0 ],
wt = ϑ F ij (t, x, y, w, Dw) wij + G(t, x, y, w, Dw)
w(·, 0) = w0
on Ω
admits a solution w in the space Csk,2+α (Ω × [0, τ0 ]). Moreover,
||w||Csk,2+α (Ω×[0,τ0 ]) ≤ C ||w0 ||Csk,2+α (Ω)
for some positive constant C which depends only on α, k, λ, ν and σ.
Proof. For any number 0 < τ ≤ T , we define the operator
M̄ : Csk,2+α (Ω × [0, τ ]) → Csk,α (Ω × [0, τ ]) × Csk,2+α (Ω)
by
M̄ w = ( M w, wt=0 ).
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
949
Our assumptions imply that M̄ is a well defined bounded operator. We will show
that if τ is sufficiently small, there exists a function w ∈ Csk,2+α (Ω × [0, τ ]) and a
function g ∈ Csk,α (Ω × [0, τ ]) such that
M̄ (w) = (g, w0 ) and g(·, t) ≡ 0,
∀t ∈ [0, τ0 ]
for some number τ0 ∈ (0, τ ], depending only on absolute constants. This will imply
that our initial value problem is solvable on Ω × [0, τ0 ], finishing the proof of the
Theorem.
As before, we denote by P (w) the operator
P (w) = ϑ F ij (t, x, y, w, Dw) wij + G(t, x, y, w, Dw).
Since P (w0 ) belongs to the space Csk,α (Ω), by Theorem I.11.7, there exists a function w̄ in the space Csk,2+α (Ω × [0, τ ]) such that
w̄ = w0
and w̄t = P (w0 )
at t = 0.
It follows that the function
ḡ = w̄t − P (w̄)
belongs to the space Csk,α (Ω × [0, τ ]) and satisfies
g(·, t) ≡ 0
at t = 0.
We will prove, using the Inverse Function Theorem, that the operator M̄ is
invertible in a small neighborhood
||g − ḡ||Csk,α (Ω×[0,τ ]) ≤ ,
||wt=0 − w0 ||Csk,α (Ω) ≤ of the point (ḡ, w0 ). For this purpose it is enough to show that the derivative of
the operator M at w̄,
DM̄ (w̄) : Csk,2+α (Ω × [0, τ ]) → Csk,α (Ω × [0, τ ]) × Csk,2+α (Ω)
defined by
DM̄ (w̄)(w̃) = ( DM (w̄)(w̃), w̃t=0 ),
is an invertible linear map. But this follows immediately from Theorem II.1.1, since
the linear operator DM (w̄) satisfies the required hypotheses. The same Theorem
shows that
||w̃||Csk,2+α (Ω×[0,τ ]) ≤ C ||DM (w̄)w̃||Csk,α (Ω×[0,τ ]) + ||w̃t=0 ||Csk,2+α (Ω)
with C a constant depending only on α, λ and ν, if τ ≤ 1. Notice that the constant
C(T ) in Theorem II.1.1 can be taken uniform in T , if T ≤ 1.
On the other hand, since ḡ(·, t) = 0 at t = 0, we can always find a function
g ∈ Csk,α (Ω × [0, τ ]) with ||g − ḡ||Csk,α (Ω×[0,τ ]) < and
g(·, t) ≡ 0,
∀t ∈ [0, τ0 ]
provided that τ0 ∈ (0, τ ] is sufficiently small, depending on . Therefore there exists
a function w ∈ C k,2+α (Ω ×[0, τ0 ]) such that M w = 0 in Ω ×[0, τ0 ] and w(·, 0) = w0 .
Moreover w will satisfy the estimate
||w||Csk,2+α (Ω×[0,τ0 ]) ≤ C ||w0 ||Csk,2+α (Ω)
with C a constant depending only α, k, µ, λ, ν.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
950
P. DASKALOPOULOS AND R. HAMILTON
Part III. The free boundary for the porous medium equation
III.1. The result. In this last part of the paper we will apply the results of Part
II to study the regularity of the free boundary for the porous medium equation
ut = ∆um
with exponent m > 1. Introducing the pressure
f = m um−1
we can express the porous medium equation in terms of f , namely
ft = f ∆f + r(m) |Df |2
with
r(m) = 1/(m − 1) > 0.
Let Ω be a compact domain in R2 and f 0 a function on Ω with f 0 = 0 at ∂Ω
and f 0 > 0 in the interior of Ω. We will study the free-boundary problem
(
(x, y, t) ∈ Ωt × [0, T ],
ft = f ∆f + r |Df |2 ,
0
f (x, y, 0) = f ,
(x, y) ∈ Ω,
for some T > 0, where Ωt is the closure of the set
{(x, y) ∈ R2 : f (x, y, t) > 0}
and r is any positive number. Here and throughout the rest of the paper we will
denote by Ωt × [0, T ] the set
[
(Ωt × {t}).
Ωt × [0, T ] =
0≤t≤T
Our main result is the following regularity theorem.
III.1.1. Theorem. Let Ω be a compact domain in R2 and let f 0 be a function in
the space Cs2+α (Ω), for some 0 < α < 1, with f 0 = 0 at ∂Ω and f 0 > 0 in the
interior of Ω. Moreover, assume that
Df 0 (x, y) 6= 0,
∀(x, y) ∈ Ω.
Then, there exists a number T > 0, for which the free-boundary problem
(
(x, y, t) ∈ Ωt × [0, T ],
ft = f ∆f + r |Df |2 ,
0
f (x, y, 0) = f ,
(x, y) ∈ Ω,
with r > 0, admits a solution f which is smooth up to the free boundary ∂Ωt ×(0, T ].
In particular, the interface ∂Ωt × (0, T ] is smooth.
The Theorem stated in the Introduction is an immediate consequence of Theorem
III.1.1, since C α (Ω) ⊂ Csα (Ω), for all α > 0. Notice that the solution of the free
boundary problem in Theorem III.1.1, extended to be equal to zero on (R2 \ Ωt ) ×
[0, T ], is nothing but the unique weak solution of the Cauchy problem
(
(x, y, t) ∈ R2 × [0, T ],
ft = f ∆f + r |Df |2 ,
f (x, y, 0) = f 0 ,
(x, y) ∈ R2 .
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
951
III.2. Local coordinate change. To motivate the proof of the regularity Theorem III.1.1 we will first compute the transformation of the equation of the pressure
ft = f ∆f + r |Df |2
when one exchanges dependent and independent variables near the boundary. This
change of coordinates converts the free boundary into a fixed boundary. More
precisely, assume for the moment that the function f belongs to the space
Cs2+α (Ωt × [0, T ]). Pick a point P0 = (x0 , y0 , t0 ) at the free boundary ∂Ωt × (0, T ].
We can assume (by rotating the coordinates) that
fx (P0 ) > 0,
ft (P0 ) = r fx2 (P0 ) > 0.
fy (P0 ) = 0,
For a positive number δ sufficiently small, we denote by Qδ the cube
|x − x0 | ≤ δ,
|y − y0 | ≤ δ,
−δ ≤ t − t0 ≤ 0,
and by Λδ , the intersection
Λδ = (Ωt × [0, T ]) ∩ Qδ .
Since the first derivatives of f are Hölder continuous up to the free boundary, there
exists a number δ > 0 such that
fy (x, y, t) ∼ 0,
fx (x, y, t) > 0,
ft (x, y, t) > 0,
∀(x, y, t) ∈ Λδ .
It follows from the Implicit Function Theorem that if the number δ is sufficiently
small, we can solve the equation z = f (x, y, t), for (x, y, t) ∈ Λδ with respect to x,
yielding to a map
x = h(z, y, t)
defined for all (z, y, t) sufficiently close to the point (0, y0 , t0 ). Also, after a moment
of thought we can see that there exists a number η > 0 sufficiently small, so that h
is defined on the cube
Bη = { 0 ≤ z ≤ η, |y − y0 | ≤ η, −η ≤ t − t0 ≤ 0 }.
We wish to find the differential equation satisfied by the function h in Bη . Notice
that the free boundary f = 0 has now been transformed into the fixed boundary
z = 0. We can easily compute the transformation of the first order derivatives by
the equations
hz fx = 1,
hz ft + ht = 0,
hz f y + hy = 0
which yield to
1
ht
hy
,
ft = − ,
fy = − .
hz
hz
hz
We can differentiate once more to compute the transformation of the second order
derivatives. We have
fx =
hzz fx2 + hz fxx = 0
and
hzz fy2 + 2 hzy fy + hyy + hz fyy = 0.
Hence
fxx = −
fx2
1
hzz = − 3 hzz
hz
hz
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
952
P. DASKALOPOULOS AND R. HAMILTON
and
fyy
1
1
=−
fy2 hzz + 2 fy hzy + hyy = −
hz
hz
h2y
hy
hzz − 2 hzy + hyy
h2z
hz
!
.
Notice that the function h belongs to the space Cs2+α (Bη ), since we have assumed
that f ∈ Cs2+α (Ωt ×[0, T ]). This follows immediately from the above computations.
Via this change of coordinates, the equation
ft − f (fxx + fyy ) − r (fx2 + fy2 ) = 0
transforms into the equation
M (h) = ht − z
1 + h2y
2hy
hzz −
hzy + hyy
h2z
hz
!
+r
1 + h2y
= 0.
hz
The operator M defined above is a quasilinear operator which becomes degenerate
when z = 0. We can easily compute its linearization DM (h) at the point h:
!
1 + h2y
2hy
h̃zz −
h̃zy + h̃yy
DM (h)(h̃) = h̃t − z
h2z
hz
!
1 + h2y
1 + h2y
z hy hzy
h̃z
− 2z
hzz + 2
− r
h2z
h3z
h2z
hy
hy
2zhzy
h̃y .
+ 2r
− 2z 2 hzz +
hz
hz
hz
Since h ∈ Cs2+α (Bη ) and on the set Bη we have
hy (x, y, t) ∼ 0
hz (z, y, t) > 0,
the linearized operator DM (h) on Bη belongs to the class of the degenerate operators studied in Part II.
Our goal is to use the results in Part II for the proof of Theorem III.1.1. However,
the change of coordinates presented here is only local and can’t be used directly for
the proof of Theorem III.1.1. In the next section we will introduce a more subtle,
global change of coordinates which is based on similar ideas.
III.3. Partial regularity. Let Ω be a compact domain in R2 and f 0 a function
on Ω with f 0 = 0 at ∂Ω and f 0 > 0 in the interior of Ω. Assume that f 0 ∈
Csk,2+α (Ω), for some nonnegative integer k and that
|D f 0 (x, y)| + f 0 (x, y) ≥ c > 0
∀(x, y) ∈ Ω
for some fixed positive number c. Denote by D the unit disk
D = {(u, v) ∈ R2 : u2 + v 2 ≤ 1 }
and pick a smooth surface S, sufficiently close to the surface z = f 0 (x, y). Let
S : D → R3 be a smooth parametrization for the surface S which maps ∂D onto
S ∩ { z = 0 }. Also, let
T1
T = T2
T3
be a smooth vector field, transverse to the surface S. Since |Df 0 | ≥ c along ∂Ω and
S is sufficiently close to the surface z = f 0 (x, y), we can choose T to be parallel
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
953
to the plane z = 0 in a small neighborhood of ∂D. In other words, there exists a
number δ > 0 depending on k such that
T3 ( uv ) = 0
on D \ D1−δ
with D1−δ = {(u, v) ∈ R2 : u2 + v 2 ≤ 1 − δ }.
For η > 0 sufficiently small, we define the change of spatial coordinates Φ :
D × [−η, η] → R3 by
 
 
x
u
u
u




y =Φ v =S
+wT
(III.3.1)
.
v
v
z
w
The map Φ defines x, y and z as functions of the new coordinates u, v and w.
Assume that z = f (x, y, t) satisfies the initial free-boundary problem
(
(x, y, t) ∈ Ωt × [0, T ],
ft = f ∆f + r |Df |2 ,
0
f (x, y, 0) = f ,
(x, y) ∈ Ω,
for some r > 0, where Ωt is the closure of the set {(x, y) ∈ R2 : f (x, y, t) > 0}.
Under the coordinate change Φ, the initial data f 0 (x, y) transforms to a function
w0 (u, v) which can be made arbitrarily small, by choosing the smooth surface S
sufficiently close to the surface z = f 0 (x, y). We will see that w0 ∈ Csk,2+α (D),
since f ∈ Csk,2+α (Ω).
When z evolves as a function of (x, y, t), then, through this coordinate change
w evolves as a function of (u, v, t) with (u, v) ∈ D. Notice that by our choice of
the parametrization S we have (u, v) ∈ ∂D iff z = 0. Hence the free-boundary
where z = f (x, y, t) = 0 is mapped onto the fixed lateral boundary of the cylinder
D × [0, T ].
The evolution of w is described in the following Theorem, where to simplify the
notation we use subscripts i, j, k ∈ {u, v} to denote differentiation with respect to
the variables u, v:
III.3.2. Theorem. Let Ω be a compact domain in R2 , k a nonnegative integer
and let f 0 be a function in the space Csk,2+α (Ω), for some number α in 0 < α < 1.
Assume that f 0 > 0 in the interior of Ω and f 0 = 0 at ∂Ω with
|D f 0 (x, y)| + f 0 (x, y) ≥ c
∀(x, y) ∈ Ω.
Then, under the coordinate change (III.3.1) the initial free boundary problem
(
(x, y, t) ∈ Ωt × [0, T ],
ft = f ∆f + r |Df |2 ,
0
f (x, y, 0) = f ,
(x, y) ∈ Ω,
with r > 0, converts into the initial value problem
(
M w = 0,
(u, v, t) ∈ D × [0, T ],
0
w(u, v, 0) = w ,
(u, v) ∈ D,
with
M w = wt − (ϑ F ij (t, u, v, w, Dw) wij + G(t, u, v, w, Dw))
and w0 ∈ Csk,2+α (D). Moreover, if T ≤ τk , with τk sufficiently small depending on
c and k, the operator M satisfies all the hypotheses of Theorem II.2.1.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
954
P. DASKALOPOULOS AND R. HAMILTON
Proof. We begin by selecting a sufficiently small number δ > 0, such that
T3 ( uv ) = 0
with T =
T1
T2
T3
on D \ D1−δ
denoting, as above, the transverse vector field to the surface
S. Notice that by choosing the smooth surface sufficiently close to the surface
z = f 0 (x, y), we can make δ depend only on the constant c.
To show that w satisfies an equation of the desired form in the interior cylinder
D1−δ × [0, T ] is straightforward. Hence, we will restrict our attention to D \ D1−δ .
We start by expressing the first and second derivatives of z with respect to x, y, t
in terms of the first and second derivatives of w with respect to u, v, t.
We begin with the first order derivatives. Since x, y and z are functions of u, v
and w, while w is a function of u, v and t we have
 ∂x
(III.3.3)

∂u
∂y
∂u
∂x
∂v
∂y
∂v


=
xu + xw ∂w
∂u
yu + yw ∂w
∂u
xw ∂w
∂v
yw ∂w
∂v
xv +
yv +


where xu , yu , zu , xv , yv , zv and xw , yw , zw , denote the partial derivatives of the functions x = x(u, v, w), y = y(u, v, w) and z = z(u, v, w) with respect to u, v, w respectively.
Therefore we can compute the partial derivatives of the functions u = u(x, y, t)
and v = v(x, y, t) by
∂x
∂v 
∂x
∂u
∂y
∂v
∂y
 ∂u
(III.3.4)

 ∂x
=
∂u
∂y
∂u
∂x
∂v
∂y
∂v

−1

=
∂y
∂v
1 
D
− ∂x
∂v
∂y
− ∂u


∂x
∂u
with
D=
∂x ∂y ∂x ∂y
−
.
∂u ∂v
∂v ∂u
Since for u2 + v 2 ≥ 1 − δ the coordinate z is independent of w because T3 = 0, we
have
 ∂z   ∂u
∂v 
zu ∂x + zv ∂x
∂x

 =
(III.3.5)
∂z
∂u
∂v
z
+
z
u ∂y
v ∂y
∂y
and hence from (III.3.4)
 ∂y
∂v
1

 =
D
∂z
− ∂x
∂y
∂v
 ∂z 
∂x
∂y
− ∂u
∂x
∂u
 
zu
  .
zv
We compute next the transformation of the second order derivatives. Differentiating the function z in (III.3.5) once more with respect to x and y and summing
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
955
up we obtain
2
2
∂2z
∂2v
∂ u ∂2u
∂ v
∂2z
+
z
+
=
z
+
+
u
v
∂x2
∂y 2
∂x2
∂y 2
∂x2
∂y 2
" #
2
2
n
∂u ∂v
∂u
∂u ∂v
∂u
+
+ 2zuv
+
+ zuu
∂x
∂y
∂x ∂x ∂y ∂y
" 2 # o
2
∂v
∂v
.
+ zvv
+
∂x
∂y
The second derivatives ∂ 2 u/∂x2 and ∂ 2 v/∂x2 satisfy the linear system

 ∂x ∂x   ∂ 2 u   ∂ 2 x ∂u 2
2
∂ 2 x ∂u ∂v
∂v 2
+ 2 ∂u∂v
+ ∂∂vx2 ∂x
2
∂u
∂v
∂x2
∂u
∂x
∂x
∂x


+

=0

2
2
2
2
∂y
∂y
2
2
∂ v
∂ y ∂u ∂v
∂ y ∂v
∂ y ∂u
+ 2 ∂u∂v ∂x ∂x + ∂v2 ∂x
∂u
∂v
∂x2
∂u2 ∂x
while the derivatives ∂ 2 u/∂y 2 and ∂ 2 v/∂y 2 satisfy a similar system. After several
simple calculations we conclude that
∂2x
∂2x
∂2z
∂2z
1
∂2x
∂y
∂y
− zu
A 2 − 2B
+C 2
zv
+ 2 = 3
∂x2
∂y
D
∂u
∂v
∂u
∂u∂v
∂v
∂2y
∂2y
∂2y
∂x
∂x
− zv
A 2 − 2B
+C 2
+ zu
∂v
∂u
∂u
∂u∂v
∂v
1
+ 2 ( Azuu − 2Bzuv + Czvv )
D
where, to simplify the notation, we have set
2 2
2 2
∂x ∂x
∂y ∂y
∂y
∂x
∂x
∂y
+
,
C=
+
,
B=
+
.
A=
∂v
∂v
∂v ∂u ∂v ∂u
∂u
∂u
On the other hand, differentiating (III.3.3) we obtain
∂w
∂2w
∂ 2x
+ xw 2 ,
= xuu + 2xuw
2
∂u
∂u
∂u
∂2x
∂w
∂2w
+ xw 2
= xvv + 2xvw
2
∂v
∂v
∂v
and
∂w
∂w
∂2w
∂2x
= xuv + xuw
+ xvw
+ xw
∂u∂v
∂v
∂u
∂u∂v
while we have similar expressions for the derivatives ∂ 2 y/∂u2 , ∂ 2 y/∂v 2 and
∂ 2 y/∂u∂v.
Using the above computations we now find that
∂2w
∂2z
1
∂2w
∂2w
∂2z
+C
+ 2 = − 3 (xw K + yw M ) A
− 2B
∂x2
∂y
D
∂u2
∂u∂v
∂v 2
∂w
2
− 3 [K(Axuw − Bxvw ) + M (Ayuw − Byvw )]
D
∂u
∂w
− [K(Cxvw − Bxuw ) + M (Cyvw − Byuw )]
∂v
1
− 3 [ K(Axuu − 2Bxuv + Cxvv ) + M (Ayuu − 2Byuv + Cyvv ) ]
D
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
956
P. DASKALOPOULOS AND R. HAMILTON
where, to simplify the notation, we have set
∂y
∂y
∂x
∂x
− zv ,
M = zv
− zu .
∂v
∂u
∂u
∂v
Notice that under this notation we have
K
∂z
M
∂z
=
and
=
.
∂x
D
∂y
D
K = zu
Finally, to compute the transformation of the time derivative ∂z/∂t, we differentiate z = f (x, y, t) with respect to time t. We find
∂z
∂z
∂z ∂w
+ xw
+ yw
=0
∂t
∂x
∂y ∂t
which gives
D
∂w
∂z
=−
.
∂t
xw K + yw M ∂t
We are now ready to compute the evolution equation of w on (D \ Dδ ) × [0, T ].
Since z = f (x, y, t) evolves as
ft = f (fxx + fyy ) + r (fx2 + fy2 )
from the above computations we conclude that w evolves as
z
∂2w
∂2w
∂2w
∂w
= 2 A
+
C
−
2B
∂t
D
∂u2
∂u∂v
∂v 2
∂w
2z
[K(Axuw − Bxvw ) + M (Ayuw − Byvw )]
+ 2
D (xw K + yw M )
∂u
∂w
+ [K(Cxvw − Bxuw ) + M (Cyvw − Byuw )]
∂v
z
+ 2
{K(Axuu − 2Bxuv + Cxvv ) + M (Ayuu − 2Byuv + Cyvv )}
D (xw K + yw M )
r
(K 2 + M 2 ).
−
D (xw K + yw M )
Next notice that on D\Dδ the z coordinate in (III.3.1) is independent of w (T3 = 0).
In other words z = S3 , with S3 denoting the z coordinate of the parametrization
S. Therefore, since S3 maps ∂D onto z = 0, we can write z = ϑ(u, v) on D \ Dδ ,
where ϑ is a smooth function, strictly positive in the interior of D and such that
ϑ(u, v) ∼ d(u, v)
on D \ D1−δ
with d(u, v) denoting the distance of the point (u, v) to the boundary of D.
We can rewrite the equation of w on (D \ D1−δ ) × [0, T ] as
ϑ
∂2w
∂2w
∂2w
∂w
= 2 A
+C
− 2B
∂t
D
∂u2
∂u∂v
∂v 2
2
2
r (K + M )
+ ϑ H(t, u, v, w, Dw).
−
D(xw K + yw M )
It is then easy to see that the equation takes the form
wt − (ϑ F ij (t, u, v, w, Dw) wij + G(t, u, v, w, Dw) ) = 0.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
957
The subindices i, j denote differentiation with respect to the variables u, v respectively. As always the summation convention is used. Under this notation
F ii =
A
,
D2
F ij = F ji = −
B
,
D2
F jj =
C
D2
and
G(t, u, v, w, Dw) = −
r (K 2 + M 2 )
+ ϑ H(t, u, v, w, Dw).
D(xw K + yw M )
The coordinate change Φ in (III.3.1) transforms the initial data f 0 (x, y) into
a function w0 (u, v). The reader can verify that if f 0 ∈ Csk,2+α (Ω), then w0 ∈
Csk,2+α (D). Moreover, the norm ||w0 ||Csk,2+α (D) can be made arbitrarily small, by
choosing the smooth surface S in (III.3.1) sufficiently close to the surface z =
f 0 (x, y).
Since the change of variables Φ in (III.3.1) is smooth, the functions F ij and
G will depend smoothly on the variables (t, u, v, w, Dw) as long as | D | > 0 and
| xw K + yw M | > 0 for all (u, v) ∈ D \ Dδ . Therefore, in order to prove that the
operator
M w = wt − (ϑ F ij (t, u, v, w, Dw) wij + G(t, u, v, w, Dw) )
satisfies the hypotheses of Theorem II.2.1 on (D \ D1−δ ) × [0, T ], for T sufficiently
small, it will be enough to show that at t = 0 and for δ small,
|D| > 0
and
| xw K + yw M | > 0
on D \ D1−δ
the matrix
F ij (t, u, v, w0 , Dw0 )t=0 ,
(u, v) ∈ D \ D1−δ ,
is strictly positive, and also
∂
−r (K 2 + M 2 )
(t, u, v, w0 , Dw0 ) > 0
nl
∂wl D(xw K + yw M ) t=0
and
ϑ
∂H
∂wl
(t, u, v, w0 , Dw0 ) = 0
at ∂D
at ∂D.
t=0
Here Dw = (wu , wv ), l ∈ {u, v} and n = (n1 , n2 ) denotes the interior normal to
∂D. Once more the summation convention is used.
Choose a point (u0 , v0 ) ∈ ∂D which is mapped, through our coordinate change
Φ, to the point (x0 , y0 ) ∈ S ∩ {z = 0}. We can assume, by rotating the (u, v)-axes,
that (u0 , v0 ) is the point (1, 0). Moreover, we can rotate the (x, y)-axes so that the
exterior normal to S ∩ {z = 0} at (x0 , y0 ) is parallel to the vector (1, 0). Therefore,
at (u0 , v0 ) = (1, 0) we have
xu ∼ 1,
xv ∼ 0,
yu ∼ 0,
yv ∼ ±1
in the coordinate change (III.3.1). Hence, for w0 ∼ 0, Dw0 ∼ 0 (which holds true
since ||w0 ||C 2+α (D) can be made arbitrarily small) we compute
 ∂x ∂x  

1 0
∂u
∂v

∼
.
∂y
∂y
0 ±1
∂u
∂v
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
958
P. DASKALOPOULOS AND R. HAMILTON
It is now easy to see that

A
B

B


∼
C
1 0


0 1
and
D ∼ yv ∼ ±1,
while
K ∼ z u yv
and
M ∼ zv .
Since the exterior normal to S ∩ {z = 0} at (x0 , y0 ) is parallel to the vector (1, 0)
and S can be chosen arbitrarily close to z = f 0 (x, y) we have
∂f 0
<0
∂x
and
∂f 0
∼0
∂y
and
M
∂z
∂f 0
=
∼
∼0
D
∂y
∂y
at the point (x0 , y0 ). Hence
∂z
∂f 0
K
=
∼
<0
D
∂x
∂x
at the point (1, 0). Therefore
K
∼ −1
D
at the point (1, 0). We conclude that
zu ∼
and
zv ∼ M ∼ 0
xw K + yw M ∼ −1
at the point (1, 0), since xw = T1 ∼ 1. Hence
|D| ∼ 1
|xw K + yw M | ∼ 1
and
at (1, 0), as desired. Moreover, the matrix
1 AB
(
)
D2 B C
is strictly positive at t = 0 at the point (1, 0).
We will show next that at t = 0 and at the point (1, 0) we have
∂
K2 + M 2
> 0.
∂wu D(xw K + yw M )
( F ij ) =
To simplify the notation we denote by Ku , Mu , Du the partial derivatives ∂K/∂wu ,
∂M/∂wu and ∂D/∂wu respectively. The reader can easlily check that xw K + yw M
is independent of wu . Hence
−(K 2 + M 2 )Du
2 (KKu + M Mu )
∂
K2 + M 2
= 2
+
∂wu D(xw K + yw M )
D (xw K + yw M )
D(xw K + yw M )
with
∂w
∂w
− yw xv + xw
= xw yv − yw xv ∼ xw yv
Du = xw yv + yw
∂v
∂v
and
Ku = −zv yw ∼ 0
and
Mu = zv xw ∼ 0,
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
959
since xv ∼ 0 and zv ∼ 0 at the point (1, 0). Finally we conclude that
K2 1
K2 + M 2
∂
∼− 2
∼1
∂wu D(xw K + yw M )
D xw zu
since K/D ∼ −1, M ∼ 0, zu ∼ −1 and xw = T1 ∼ 1 at (1, 0). We conclude that
∂
−r (K 2 + M 2 )
∼1
nl
∂wl D(xw K + yw M )
at the point (1, 0), since the exterior normal n = (n1 , n2 ) at (1, 0) is (1, 0).
It remains to be shown that ϑ(∂H/∂wl ) = 0 at (t, u, v, w0 , Dw0 ) for t = 0 and
(u, v) ∈ ∂D. Since the coordinate change Φ is smooth and the function w0 belongs
to the space Csk,2+α (D) while
ϑ(u, v) ∼ d(u, v)
as (u, v) → ∂D
with d(u, v) denoting the distance to the boundary of D, this follows as an immediate consequence from the next simple lemma.
III.3.6. Lemma. Let ξ be a function in Csk,2+α (D), for some nonnegative integer
k and some number α in 0 < α < 1. Then
d D2 ξ = 0
at ∂D
with d denoting the distance function to the boundary of D.
Proof of Lemma. The lemma is obvious when k > 1. Hence we can assume that
ξ ∈ Cs2+α (D). Pick a point (u0 , v0 ) ∈ ∂D, wich can be taken without loss of
generality to be the point (1, 0). We will show that d ξuu , d ξuv and d ξvv are all
equal to zero at (1, 0). We proceed by contradiction. Assume that d ξuu = α > 0
at (1, 0) (if α < 0 we replace ξ by −ξ). Then by continuity
α
>0
ξuu (u, 0) ≥
2(1 − u)
for all 1 − ≤ u ≤ 1, for some > 0. Hence, for 1 − < u1 < u2 < 1, we have
Z u2
Z u2
α
1 − u1
α
du ≥ ln
ξuu (u, 0) du ≥
.
ξu (u2 , 0) − ξu (u1 , 0) =
2
1 − u2
u1
u1 2(1 − u)
Letting u2 → 1 we derive a contradiction, since ξu is bounded on D. Assume next
that d ξvv = α > 0 at (0, 1). Then, by continuity
α
>0
ξvv (u, v) ≥
2d
√
for all (u, v) ∈ D with 1 − 2 ≤ u ≤ 1 and − ≤ v ≤ , for some > 0. Hence
Z /2
p
p
p
α
ξvv ( 1 − 2 , v) dv ≥ .
ξv ( 1 − 2 , /2) − ξv ( 1 − 2 , −/2) =
−/2
Letting → 0 we derive a contradiction. At last the equality d ξuv = 0 at (1, 0) can
be proved in a similar manner. This finishes the proof of the lemma and also the
proof of Theorem III.3.2.
As a consequence of Theorem III.3.2 and Theorem II.2.1, we obtain the following
main result.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
960
P. DASKALOPOULOS AND R. HAMILTON
III.3.7. Theorem. Let Ω be a compact domain in R2 and let f 0 be a function
in the space Csk,2+α (Ω), for some nonnegative integer k and some number α in
0 < α < 1. Assume that f 0 = 0 at ∂Ω, f 0 > 0 in the interior of Ω so that
|D f 0 (x, y)| + f 0 (x, y) ≥ c > 0
∀(x, y) ∈ Ω.
Then, there exists a number τk , for which the initial free-boundary problem
(
(x, y, t) ∈ Ωt × [0, τk ],
ft = f ∆f + r |Df |2 ,
0
f (x, y, 0) = f ,
(x, y) ∈ Ω,
with r > 0, admits a solution f in Csk,2+α (Ωt × [0, τk ]).
Proof. As we have seen in Theorem III.3.2 the coordinate change Φ in (III.3.1)
converts the given initial free-boundary problem to the initial value problem with
fixed boundary
(
M w = 0,
(u, v, t) ∈ D × [0, T ],
0
(u, v) ∈ D,
w(u, v, 0) = w ,
with
M w = wt − ϑ F ij (t, u, v, w, Dw) wij − G(t, u, v, w, Dw)
and w0 ∈ Csk,2+α (D). However we have shown that if T ≤ τk , with τk sufficiently
small depending on k and c, the operator M satisfies all the hypotheses of Theorem 2.1 in Part II. Therefore, by this theorem the initial value problem for the
equation M w = 0 admits a solution w ∈ Csk,2+α (D × [0, τk ]). If we express the
function w in the old coordinates we obtain a solution z = f (x, y, t) of the given
initial free-boundary problem which belongs to the space Csk,2+α (Ω × [0, τk ]). The
computations are the same as in the proof of Theorem III.3.2.
III.4. C ∞ -regularity. In this final section we will give the proof of Theorem
III.1.1. It will follow from the next regularity result.
III.4.1. Theorem. Assume that for some T > 0 and some number α in 0 < α <
1, f ∈ Cs2+α (Ωt × [0, T ]) is a solution of the free-boundary problem
(
(x, y, t) ∈ Ωt × [0, T ],
ft = f ∆f + r |Df |2 ,
f (x, y, 0) = f 0 ,
(x, y) ∈ Ω,
so that
|Df (x, y, t)| + f (x, y, t) ≥ c > 0
∀(x, y, t) ∈ Ωt × [0, T ].
Then, for any positive integer k, f ∈ Csk,2+α (Ωt ×(0, T ]) and for any τ in 0 < τ < T
we have
||f ||Csk,2+α (Ωt ×[τ,T ]) ≤ Ck (τ, ||f 0 ||Cs2+α (Ω) ).
Proof. We begin with the proof of the theorem for k = 1. Choose a family of
smooth regularizations f0 of the initial data f 0 , approximating f 0 , so that each f0
is strictly positive in the interior of a compact domain Ω , with f = 0 at ∂Ω and
|Df0 (x, y)| + f0 (x, y, t) ≥ c > 0
∀(x, y) ∈ Ω .
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
961
It follows from the results in Section 11, Part I, that we can choose the functions
f0 so that
||f0 ||Cs2+α (Ω ) ≤ C ||f 0 ||Cs2+α (Ω)
with the constant C independent of and f0 → f 0 as → 0 in the sense that
Dxi Dyj Dtk f → Dxi Dyj Dtk f
if j + j + k ≤ 1 and
d Dxi Dyj f → d Dxi Dyj f
if j + j = 2, with d denoting the distance to the boundary of Ω .
Denote by f the solution to the free boundary problem
(
ft = f ∆f + r |Df |2 ,
(x, y, t) ∈ Ω,t × [0, τ1 ],
0
f (x, y, 0) = f ,
(x, y) ∈ Ω ,
as constructed in Theorem III.3.7. It follows from the same theorem that there
exists a number τ1 in 0 < τ1 ≤ T which is independent of such that f ∈
Cs1,2+α (Ω,t × [0, τ1 ]). Moreover,
||f ||Cs1,2+α (Ω,t ×[0,τ1 ]) ≤ C(||f0 ||Cs1,2+α (Ω ) ).
On the other hand, since ||f0 ||Cs2+α (Ω ) ≤ C ||f 0 ||Cs2+α (Ω) the result in Theorem
III.3.7 implies that
||f ||Cs2+α (Ω,t ×[0,T ]) ≤ C(||f 0 ||Cs2+α (Ω) )
which is a bound that is independent of . Therefore, it is easy to see that passing
to a subsequence (still denoted by f ) the sequence f will converge to f , being the
unique weak solution of the equation
ft = f ∆f + r |Df |2
on R2 × [0, T ]
with initial data f 0 .
Our first goal is to show that for τ in 0 < τ < τ1 we have
||f ||Cs1,2+α (Ω,t ×[τ,τ1 ]) ≤ C(τ, ||f 0 ||Cs2+α (Ω) )
with C = C(||f 0 ||Cs2+α (Ω) ) independent of . For this purpose we will use the local
change of variables introduced in Section 2 of Part III.
For > 0 and τ in 0 < τ < τ1 fixed, we pick a point P0 = (x0 , y0 , t0 ) at the free
boundary ∂Ω,t × (τ, τ1 ). We can assume with no loss of generality (by rotating the
coordinates) that
(f )x (P0 ) > 0,
(f )y (P0 ) = 0,
(f )t (P0 ) = r (f )2x (P0 ) > 0.
Then, as we showed in Section 2, locally around the point P0 , we can solve the
equation z = f (x, y, t) with respect to x yielding to a function x = h (z, y, t)
defined on a small box
Bη = { 0 ≤ z ≤ η, |y − y0 | ≤ η, −η ≤ t − t0 ≤ 0 }.
After a moment of thought we realize that the number η can be taken to be independent of the particular point P0 on the free boundary Ω,t × [0, T ] and independent
. This is because the norms ||f ||Cs2+α (Ω ×[0,T ]) are uniformly bounded.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
962
P. DASKALOPOULOS AND R. HAMILTON
To simplify the notation we will denote for the moment the function h by h.
We have computed in Section 2 that on the set Bη we have
hz =
1
,
fx
hy = −
fy
hz
and therefore
hz (z, y, t) > 0,
hy (x, y, t) ∼ 0
if η is sufficiently small. Moreover, each h satisfies the equation
!
)
(
1 + h2y
1 + h2y
2hy
ht − z
=0
hzz −
hzy + hyy − r
h2z
hz
hz
for all (z, y, t) ∈ Bη .
Our goal is to establish a Cs1,2+α bound for h = h in the box Bη which is
independent of . To do so we will compute, by differentiating the above equation,
the evolution of the derivatives hz , hy and ht on Bη and then we will apply the
local Shauder estimates in Theorem II.1.2.
We will first establish this bound for hy . The reader can easily chack that w = hy ,
when restricted on the box Bη , satisfies the equation M,y (w) = 0 where
!
n
1 + h2y
2hy
wzz −
wzy + wyy
M,y (w) = wt − z
h2z
hz
!
r(1 + h2y ) 2z(1 + h2y )hzz
2zhy hzy
wz
−
+
+
h2z
h3z
h2z
o
2zhy hzz
2zhzy
−2rhy
wy .
+
+
−
hz
h2z
hz
After a moment of thought we see that all the coefficients of the operator M,y
belong to the space Csα (Bη ) with norms uniformly bounded by a constant which
depends only on ||f 0 ||Cs2+α (Ω) . Moreover,
r(1 + h2y ) 2z(1 + h2y )hzz
2zhy hzy
r
−
+
≥ >0
2
3
2
hz
hz
hz
2
if η is sufficiently small, independently of . Therefore, the operator M,y satisfies
all the hypotheses of Theorem 1.3 in Part II and in conclusion the following estimate
holds (for h = h ):
khy kCs2+α (Bη/2 ) ≤ C(||f 0 ||Cs2+α (Ω) ).
The same estimate can be shown to hold true for the time derivative w = ht which
satisfies the same equation, namely M,t (w) = 0, with
!
n
1 + h2y
2hy
wzz −
wzy + wyy
M,t (w) = wt − z
h2z
hz
!
r(1 + h2y ) 2z(1 + h2y )hzz
2zhy hzy
wz
−
+
+
h2z
h3z
h2z
o
2zhy hzz
2zhzy
−2rhy
wy .
+
+
−
hz
h2z
hz
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
963
Finally, the derivative w = hz satisfies the equation M,z = 0 where M,z is the
operator
!
n
1 + h2y
2hy
M,z (w) = wt − z
wzz −
wzy + wyy
h2z
hz
!
(r + 1)(1 + h2y ) 2z(1 + h2y )hzz
wz
−
+
h2z
h3z
o
2(r + 1)hy
4zhy hzz
2zhzy
wy + hyy .
+ −
+
−
2
hz
hz
hz
Since h ∈ Cs1,2+α (Bη/2 ), the function hyy belongs in Cs1,2+α (Bη/2 ). Hence, the
desired estimate follows as above from Theorem 1.3 in Part II.
We conclude from the above that h = h satisfies
kh kCs1,2+α (Bη/2 ) ≤ C(||f 0 ||Cs2+α (Ω) ).
We can now go back to the original coordinates. The transformation of first and
second derivatives via the coordinate change, as computed in Section 2, and our
estimates above show that if for some number τ in 0 < τ < τ1 , Aτη, denotes the set
Aτη, = { (x, y, t) ∈ Ω × [τ, τ1 ] : f (x, y, t) ≤ η },
then we have
||f ||Cs1,2+α (Aτ
η, )
≤ C(τ, ||f 0 ||Cs2+α (Ω) ).
The same estimate holds true on the interior set
Γτη, = (Ω,t × [τ, τ1 ]) \ Aτη,
as it follows from the standard regularity theory of nondegenerate parabolic equations. Therefore we conclude that
||f ||Cs1,2+α (Ω ×[τ,τ1 ]) ≤ C(τ, ||f 0 ||Cs2+α (Ω) )
for all τ in 0 < τ < τ1 , as desired.
Next we extend each solution f so that f (x, t) = 0 for all x ∈ R2 \ Ω,t ,
0 ≤ t ≤ τ1 , so that each f becomes a weak solution of the Cauchy problem
(
(x, y, t) ∈ R2 × [0, τ1 ],
ft = f ∆f + r |Df |2 ,
f (x, y, 0) = f0 ,
(x, y) ∈ R2 .
The sequence of solutions {f } is equicontinuous and uniformly bounded on R2 ×
[0, τ1 ], since
||f ||Cs2+α (Ω ×[0,τ1 ]) ≤ C(||f 0 ||Cs2+α (Ω) ).
Therefore there exists a subsequence, still denoted by f which converges, as → 0,
to the given solution f (extended to be equal to zero outside Ωt × [0, τ1 ]). The
convergence is uniform on compact subsets of R2 × [0, τ1 ]. In particular
dist (∂Ω,t , ∂Ωt ) → 0
as → 0
uniformly on t in the interval 0 ≤ t ≤ τ1 . However, when restricted on Ωt × [τ, τ1 ],
for any τ in 0 < τ < τ1 the convergence is much stronger. Indeed, it follows from
the uniform bound
||f ||Cs1,2+α (Ω ×[τ,τ1 ]) ≤ C(τ, ||f 0 ||Cs2+α (Ω) )
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
964
P. DASKALOPOULOS AND R. HAMILTON
that there exists a subsequence of {f }, still denoted by f , such that for any
(x, t) ∈ Ω × [τ, τ1 ] we have
Dxi Dyj Dtk f (x, y, t) → Dxi Dyj Dtk f (x, y, t)
if i + j + k ≤ 2 and
d Dxi Dyj f (x, y, t) → d Dxi Dyj f (x, y, t)
for i + j = 3, as tends to zero, where d denotes the distance to the free boundary
Ω × [τ, τ1 ]. It is now easy to check that the solution f belongs to the space
Cs1,2+α (Ω × (0, τ1 ]) and for all τ in 0 < τ < τ1 satisfies the estimate
||f ||Cs1,2+α (Ωt ×[τ,τ1 ]) ≤ C(τ, ||f 0 ||Cs2+α (Ω) ).
Finally, assume that N τ1 < T ≤ (N + 1)τ1 , for some nonnegative integer N . Since
||f ||Cs2+α (Ωt ×[0,T ]) ≤ C(||f 0 ||Cs2+α (Ω) )
we can repeat the above estimate N + 1 times to finally conclude that
||f ||Cs1,2+α (Ωt ×[τ,T ]) ≤ C(τ, ||f 0 ||Cs2+α (Ω) )
for all 0 < τ < T , as desired.
This proves the Theorem in the case of k = 1. The case of a general k can be
shown via induction by differentiating the local equation
!
1 + h2y
1 + h2y
2hy
ht − z
hzz −
hzy + hyy + r
=0
2
hz
hz
hz
k times and using once more the local Schauder estimate, Theorem II.1.3.
We finish with the proof of Theorem III.1.1.
Proof of Theorem III.1.1. From Theorem III.3.7, there exists a solution f ∈
Cs2+α (Ωt × [0, T ]) of the free boundary problem
(
(x, y, t) ∈ Ωt × [0, T ],
ft = f ∆f + r |Df |2 ,
f (x, y, 0) = f 0 ,
(x, y) ∈ Ω,
for some number T > 0. Moreover, since Df 0 6= 0 at ∂Ω, we can choose the number
T so that
|Df (x, y, t)| + f (x, y, t) ≥ c
∀(x, y, t) ∈ Ωt × [0, T ]
for some c > 0. But then, it follows from Theorem III.4.1 that
f ∈ Csk,2+α (Ωt × [0, T ]),
for all positive integers k. We conclude that f ∈ C ∞ (Ωt × [0, T ]), as desired. In
particular the free-boundary is smooth.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REGULARITY OF THE FREE BOUNDARY
965
References
S. Angenent, Analyticity of the interface of the porous media equation after the waiting
time, Proc. Amer. Math. Soc. 102 (1988), N2, 329–336 MR 89f:35103
[A1]
D.G. Aronson, Regularity properties of flows through porous media, SIAM J. Appl. Math.
17 (1969), 461–467. MR 40:571
[A2]
D.G. Aronson, Regularity properties of flows through porous media: A counterexample,
SIAM J. Appl. Math. 19 (1970), 299–307. MR 42:683
[A3]
D.G. Aronson, Regularity properties of flows through porous media: The interface, Arch.
Rational Mech. Anal. 37 (1970), 1–10. MR 41:656
[ACV] D.G. Aronson, L.A. Caffarelli, J.L. Vázquez, Interfaces with a corner-point in onedimensional porous medium flow, Comm. Pure and Appl. Math. 38 (1985), 375-404.
MR 86h:35070
[AV]
D. Aronson, J.L. Vázquez, Eventual regularity and concavity for flows in one dimensional
porous media, Arch. Rational Mech. Anal. 99 (1987), 329–348. MR 89d:35081
[C]
L. Caffarelli, Interior a priori estimates for solutions of fully non-linear equations, Ann.
of Math. 130 (1989), 189–213. MR 90i:35046
[CF1] L.A. Caffarelli and A. Friedman, Regularity of the free boundary for the one-dimensional
flow of gas in a porous medium, Amer. J. Math. 101 (1979), 1193–1218. MR 80k:76072
[CF2] L.A. Caffarelli and A. Friedman, Regularity of the free boundary of a gas flow in an
n-dimensional porous medium, Ind. Univ. Math. J. 29 (1980), 361–391. MR 82a:35096
[CVW] L.A. Caffarelli, J.L. Vázquez, N.I. Wolanski, Lipschitz continuity of solutions and interfaces of the n-dimensional porous medium equation, Ind. Univ. Math. J. 36 (1987),
373–401. MR 88k:35221
[CW] L.A. Caffarelli, N.I. Wolanski, C 1,α regularity of the free boundary for the n-dimensional
porous media equation, Comm. Pure and Appl. Math. 43 (1990), 885–902. MR 91h:35332
[HK]
K. Höllig, H.O. Kreiss, C ∞ regularity for the porous medium equation, Univ. of Winsconsin Madison, Computer Scienced Dept., Technical report # 600.
[K]
B. Knerr, The porous medium equation in one dimension, Trans. Amer. Math. Soc. 234
(1977), 381–415. MR 58:11917
[KN]
J.J. Kohn, L. Nirenberg, Degenerate elliptic-parabolic equations of second order, Comm.
Pure and Appl. Math. 20 (1967), 797–872. MR 38:2437
[S1]
M. Safonov On the classical solution of Bellman’s elliptic equations, Dokl. Akad. Nauk
SSSR 278 (1984), N4, 810–813 = Soviet Math. Dokl. 30 (1984), N2, 482–485 MR
86f:35081
[S2]
M. Safonov On the classical solution of nonlinear elliptic equations of second order, Izv.
Akad. Nauk SSSR. Ser. Mat. 52 (1988), N6, 1272–1287 = Math. USSR Izvestiya, 33
(1989), N3, 597–612 MR 90d:35104
[W1]
L. Wang On the regularity theory of fully nonlinear parabolic equations I, Comm. Pure
and Appl. Math., 45, 1992, N1, 27–76. MR 92m:35126
[W2]
L.Wang On the regularity theory of fully nonlinear parabolic equations II, Comm. Pure
and Appl. Math. 45, 1992, N2, 141–178. MR 92m:35127
[A]
Department of Mathematics, University of California, Irvine, California 92697-3875
E-mail address: [email protected]
Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0001
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use