The Classical Obstacle problem with coefficients in fractional
Sobolev spaces
Francesco Geraci
February 1, 2016
Abstract
We prove quasi-monotonicity formulas for classical obstacle-type problems with quadratic
energies with coefficients in fractional Sobolev spaces, and a linear term with a type-Dini continuity
property. Those formulas are used to obtain the regularity of free-boundary points following the
approaches by Caffarelli, Monneau and Weiss.
1
Introduction
The motivation for studying obstacle problems has roots in many applications like in physics and in
mechanics; in this regard it is possible to see [9, 17, 21, 22, 31] as some examples.
The classical obstacle problem consists in finding the minimizer of Dirichlet energy in a domain Ω,
among all functions v, with fixed boundary data, constrained to lie above a given obstacle ψ, studying
proprieties of minimizer and analysing the regularity of the boundary of the coincidence set between
minimizer and obstacle.
In the 60s it was introduced the obstacle problem within the study of variational inequalities.
In the last fifty years much efforts have been made in the understanding of the problem, a wide
variety of issues has been analysed and new mathematics ideas have been introduced. Caffarelli in [5]
introduced the so-called method of blow-up, imported by geometric measure theory for the study of
minimal surfaces, to prove some local properties of solutions. Alt-Caffarelli-Friedman [1], Weiss [35]
and Monneau [27] introduced monotonicity formulas to show the blow-up property and to obtain the
free-boundary regularity in various problems. See [4, 7, 17, 22, 30, 31] for more detailed references and
historical developments.
Recently many authors have improved classical results replacing the Dirichlet energy with a more
general variational functional and weakening the regularity of obstacle (we can see [8,12–15,26,28,33,34]).
In this context, we aim at in minimizing the following energy
ˆ
E(v) :=
hA(x)∇v(x), ∇v(x)i + 2f (x)v(x) dx,
(1.1)
Ω
among all positive functions with fix boundary data, where Ω ⊂ Rn is a smooth, bounded and open
set, n ≥ 2, A : Ω → Rn×n is a matrix-valued field and f : Ω → R is a function satisfying:
(H1) A ∈ W 1+s,p (Ω; Rn×n ) with s >
1
p
and p >
n2
n(1+s)−1
∧ n;
(H2) A(x) = (aij (x))i,j=1,...,n symmetric, continuous and coercive, that is aij = aji Ln a.e. Ω and for
some Λ ≥ 1 i.e.
Λ−1 |ξ|2 ≤ hA(x)ξ, ξi ≤ Λ|ξ|2
Ln a.e. Ω, ∀ξ ∈ Rn ;
(1.2)
(H3) f Dini-continuous, that is ω(t) = sup|x−y|≤t |f (x) − f (y)| modulus of continuity f satisfying the
following integrability condition
ˆ 1
ω(t)
dt < ∞,
(1.3)
t
0
and exists c0 > 0 such that f ≥ c0 .
Of note that we are reduced to the 0 obstacle case, so f = div(A∇ψ).
In [15] Focardi, Gelli and Spadaro study the case where A ∈ W 1,∞ (Ω). As we will see in Corollary
2.3 W 1+s,p ,→ W 1,∞ if ps > n. Consequently we assume sp ≤ n and we obtain an original result if
1
2
2
n
n
p > n(1+s)−1
∧ n (we can observe that ( n(1+s)−1
∧ n) < ns for all s ∈ R). In Remark 3.7 we will justify
the chooice in hypotesys (H1).
In the article we analyse the property of minimizers and, studying the property of blow-up, we
obtain the regularity of free-boundary.
In particular, in section 2 we fix the notation of fractional Sobolev spaces and in Proposition 2.7
prove that the minimizer u is a regulat solution of an elliptic differential equation in divergence form.
In section 3 we fix x0 point of free-boundary Γu = ∂{u = 0} ∩ Ω and by a suitable change of variable,
without loss of generality, we suppose
x0 = 0 ∈ Γu ,
A(0) = In ,
f (0) = 1.
(1.4)
We freeze argument in 0 in a way to reduce by (1.4) own problem to one with more regular coefficients.
We proceed, as in [15], introducing the rescaled 2-homogeneous functions ur (x) and their associated
energy “à la Weiss”
ˆ
ˆ
x x
u2r (x) dH n−1 (x),
Φ(r) :=
hA(rx)∇ur (x), ∇ur (x)i + 2f (rx)ur (x) dx +
A(rx) ,
|x| |x|
B1
∂B1
and prove quasi-monotonicity formulas:
Theorem 1.1. Assume that (H1)-(H3) and (1.4) are satisfied, and let Θ = Θ(n, p, s) be an exponent
(defined in (3.22)) such that Θ > n.
(i) There exist nonnegative constants C¯3 and C4 independent from r such that the function
ˆ r
n
1− n
n
ω(t) C¯3 t1− Θ(s)
− Θ(s)
C¯3 r Θ(s)
+ C4
t
+
dt
r 7→ Φ(r) e
e
t
0
is nondecreasing on (0, 12 dist(0, ∂Ω) ∧ 1).
(ii) Let 0 ∈ Sing(u) (i.e. (3.45) holds), and v be a 2-homogeneous polynomial positive function,
solving ∆v = 1 on Rn . Then, there exists a positive constant C5 independent of r such that
ˆ
n
r 7−→
(ur − v)2 dH n−1 + C5 r(1− Θ(s) ) + ω(r)
∂B1
is nondecreasing on (0, 21 dist(0, ∂Ω) ∧ 1).
In the section 4 with the help of quasi-monotonicity formulas, the non degeneracy of solution
according to Blank and Hao [2] and thanks to Γ-convergence argument we classify the blow-ups type
and so we distinguish points in Γu in regular and singular (respectively Reg(u) and Sing(u), see
Definition 4.8). Therefore following the energetic approach by Focardi, Gelli and Spadaro [15] we prove
the uniqueness of blow-up for regular and singular case. In the classical framework, the uniqueness
of the blow-ups can be derived a posteriori from the regularity properties of the free boundary (see
Caffarelli [5]). In our case, we assume hypotheses (H1)-(H3) and working as in [15] we prove the
uniqueness of blow-ups in singular points case and to obtain a uniform decay estimate for all points
in a compact subset of Sing(u). Instead, in regular points case we need to introduce an assumption
probably technical on the modulus of continuity of f (this is stronger than double Dini continuity,
see [28, Definition 1.1]):
(H3)0 Let ω(t) = sup|x−y|≤t |f (x) − f (y)| be the modulus of continuity of f and set a > 2 it holds the
following condition of integrability
ˆ 1
ω(r)
| log r|a dr < ∞.
(1.5)
r
0
So thanks to the epiperimetric inequality of Weiss [35] we obtain a uniform decay estimate for
convergence of rescaled functions to their blow-ups limit. We recall that Weiss [35] proves the
uniqueness for regular points in A = In and f = 1. Focardi, Gelli and Spadaro [15] prove our same
result for A Lipschitz continuous and f Hölder continuous. Monneau [28] proves the uniqueness of
blow-ups both for regular points and for singular points with A = In and f with Dini continuous
modulus of mean oscillation in Lp . Therefore, without further hypotheses in regular case and adding
double Dini continuity modulus of mean oscillation, Monneau gives a very accurate pointwise estimates
of decay, providing explicit form of modulus of continuity of decay.
These results allow to prove the regularity of free-boundary:
2
Theorem 1.2. We assume the hypothesis (H1)-(H3). The free-boundary decomposes as Γu =
Reg(u) ∪ Sing(u) with Reg(u) ∩ Sing(u) = ∅.
(i) Assume (H3)0 . Reg(u) is relatively open in ∂{u = 0} and for every point x0 ∈ Reg(u). there
exists r = r(x0 ) > 0 such that Γu ∩ Br (x0 ) is a C 1 hypersurface with normal versor ς is absolutely
continuous with a modulus of continuity depending on ρ defined in (4.19).
In particular if f is Hölder continuous there exists r = r(x0 ) > 0 such that Γu ∩ Br (x) is C 1,β
hypersurface for some universal exponent β ∈ (0, 1).
(ii) Sing(u) = ∪n−1
k=0 Sk and for all x ∈ Sk there exists r such that Sk ∩ Br (x) is contained in a regular
k-dimensional submanifold of Rn .
2
2.1
Preliminaries
Fractional Sobolev spaces
In order to fix the notation we report the definition of fractional Sobolev spaces. See [10, 24] for more
detailed references.
Definition 2.1. For any real λ ∈ (0, 1) and for all p ∈ [0, ∞) we define the space
|v(x) − v(y)|
p
W λ,p (Ω) := v ∈ Lp (Ω) :
∈
L
(Ω
×
Ω)
,
n
|x − y| p +λ
(2.1)
i.e, an intermediary Banach space between Lp (Ω) and W 1,p (Ω), endowed with the norm
¨
ˆ
|v|p dx +
kvkW λ,p (Ω) =
Ω
Ω×Ω
|v(x) − v(y)|p
dx dy
|x − y|n+λp
p1
.
If λ > 1 and not integer we indicate with bλc its integer part and with σ = λ − bλc its fractional part.
In this case the space W λ,p consists of functions u ∈ W bλc,p such that the distributional derivatives
Dα v ∈ W σ,p with |α| = bλc
|Dα v(x) − Dα v(y)|
p
W λ,p (Ω) := v ∈ W bλc,p (Ω) :
∈
L
(Ω
×
Ω),
∀α
such
that
|α|
=
m
.
n
|x − y| p +σ
W λ,p (Ω) is a Banach space with the norm
 p1

kvkW λ,p (Ω) = kvkpW bλc,p (Ω) +
X
kDα vkpW σ,p (Ω)  .
|α|=bλc
We state three results on Sobolev fractionary spaces useful for following. Theorems 2.2 and 2.4 are
proved, respectively in [29] and [32], for Besov spaces; thanks to [32, Remark 3.6] and [23, Theorem
14.40] we can reformulate this resuls in our notations. Theorem 2.3 is obtained combining classical
Morrey theorem, [10, Theore 8.3] and Theorem 2.2.
Theorem 2.2 ( [29, Theorem 9] and [10, Theorem 6.5]). Let Let v ∈ W λ,p (Ω) with λ > 0, 1 < p < ∞,
pλ < n and Ω ⊂ Rn be an open set of class C 0,1 with bounded boundary. Then for all 0 < t < λ there
exists a constant C = C(n, λ, p, t, Ω) for which
kvk
W
t,
np
n−(λ−t)p
(Ω)
≤ C kvkW λ,p (Ω) .
If t = 0 there exists a constant C = C(n, λ, p, Ω) for which
kvk
np
L n−λp (Ω)
≤ C kvkW λ,p (Ω) .
Theorem 2.3. Let p ∈ [1, ∞) such that sp > n and Ω ⊂ Rn be an extension domain for W λ,p . Then
there exists a positive constant C = C(n, p, λ, Ω), for which
kf kC h,α ≤ Ckf kW λ,p (Ω) ,
for all f ∈ Lp (Ω) for some α ∈ (0, 1) and h integer with h ≤ m.
3
(2.2)
1
p
and U a bounded C k domain,
γ0 : W λ,p (U ) −→ W λ− p ,p (∂U ; H n−1 ),
(2.3)
Theorem 2.4 ( [32, Theorem 3.16]). Let n ≥ 2, 0 < p < ∞, λ >
k > λ in Rn . Then there exists a bounded operator
1
such that γ0 (f ) = f|∂U for all functions f ∈ W λ,p (U ) ∩ C(U ).
Remark 2.5. Let p, λ be exponents as in theorem 2.4, pλ < n and σ := λ − bλc. If U = Br , we see how
the constant of the trace operator changes when the radius r changes.
By taking into account theorem 2.4 and 2.2 we have the following immersions
1
W λ,p (Br ) ,→ W λ− p ,p (∂Br ; H n−1 ) ,→ L1 (∂Br ; H n−1 ).
Then,setting vr (y) = v(ry)
ˆ
ˆ
n−1 y=rx n−1
|γ0 (v)(x)| dH
= r
∂Br
≤ c(1)r
n−1
X ˆ
|α|≤bλc
x= y
r
= c(1)r
n−1 − n
p
r
≤ Crn−1 r
¨
α
p
|D vr (y)| dx +
B1 ×B1
B1
X ˆ
|α|≤bλc
−n
p
|γ0 (vr )(y)| dH n−1
∂B1
|vr (y) − vr (z)|p
dy dz
|y − z|n+σp
¨
α
p
|D v(x)| dx + r
σp
Br ×Br
Br
p1
|v(x) − v(w)|p
dx dw
|x − w|n+σp
p1
kvkW λ,p (Br ) .
For which
n
kγ0 (v)kL1 (∂Br ;H n−1 ) ≤ C rn−1 r− p kvkW λ,p (Br ) .
n−λp
If p ≤ n, let n−(λ−t)p
< t < λ, i.e. (n−1)p
< t < λ, of note that λ >
np
We infer by Theorem 2.2 and by Theorem 2.4 the following
np
W λ,p (Br ) ,→ W t, n−(λ−t)p (Br ) ,→ W t−
n−(λ−t)p
np
, n−(λ−t)p
np
(2.4)
n−λp
(n−1)p
if and only if λ > p1 .
(∂Br ; H n−1 ) ,→ L1 (∂Br ; H n−1 ).
Applying the same reasoning to deduce (2.4), in particular we achieve
kγ0 (v)kL1 (∂Br ;H n−1 ) ≤ C rn−1 r−
2.2
n−(λ−t)p
p
kvk
W
t,
np
n−(λ−t)p
(Br )
.
(2.5)
Notation and first results
Let Ω ⊂ Rn be a smooth, bounded and open set, n ≥ 2, let A : Ω → Rn×n be a matrix-valued field
and f : Ω → R be a function satisfying (H1)-(H3).
Remark 2.6. By (1.2), we immediately deduce that A is bounded. In particular, kAkL∞ (Ω) ≤ Λ.
We define, for every open A ⊂ Ω and for each function v ∈ H 1 (Ω), the following energy:
ˆ
E[v, A] :=
(hA(x)∇v(x), ∇v(x)i + 2f (x)u(x)) dx,
(2.6)
A
with E[v, Ω] := E[v], and we study the relative minimum problem
inf E[·],
K
(2.7)
where K ⊂ H 1 (Ω) is the weakly closed convex given by
K := {v ∈ H 1 (Ω) | v ≥ 0 Ln -a.e. on Ω, γ0 (v) = g on ∂Ω},
1
2
(2.8)
with g ∈ H (∂Ω) being a nonnegative function. It’s easy to prove that the minimum problem (2.7)
admits a unique solution. The hypotheses (H1)-(H3) involve that the energy E is coercive and strictly
convex in K, so E is semicontinuous for the weak topology in H 1 (Ω), then there exists a unique
minimizer.
In the following we will indicate the minimum with u.
Let’s now consider the functional
ˆ
G[v] :=
hA(x)∇v(x), ∇v(x)i + 2 f (x)v + (x) dx,
(2.9)
Ω
4
1
defined on H 1 (Ω). If g̃ ∈ H 1 (Ω) and Tr(g̃) = g ∈ H 2 (∂Ω) we have
min E[·] =
K
min G[·].
(2.10)
g̃+H01 (Ω)
As in the classical case the minimum u satisfies partial differential equation both in the distributional
sense and a.e. on Ω and shows good property of regularity.
Proposition 2.7. Let u be the minimum of E in K. Then
div(A(x)∇u(x)) = f (x)χ{u>0} (x)
Therefore, if ps < n, called p∗ (s, p) := p∗ =
n
a.e. on Ω and in D0 (Ω).
(2.11)
n
∗
we have u ∈ W 2,p ∩ C 1,1− p∗ (Ω), while if ps = n
np
n−sp ,
we have u ∈ W 2,q ∩ C 1,1− q (Ω) for all 1 < q < ∞.
Proof. For the first part of proof we refer to [15, Proposition 2.2].
∗
We prove now the regularity if ps < n. By Theorem 2.2 W 1+s,p (Ω) ,→ W 1,p (Ω) with p∗ =
We also note that by the hypothesis (H1) p∗ > n, so by Morrey theorem A ∈ C
the solution of (2.11) and by theorem [20, Theorem 3.13] u ∈
1,1− n∗
Cloc p
0,1− pn∗
np
n−sp .
(Ω). Since u is
(Ω). We consider the equation
A : ∇2 v = f χ{u>0} − div(A)∇u =: g,
(2.12)
where div(A) denotes the vector with components the divergences of the columns of A. As ∇u ∈ L∞
loc (Ω)
∗
∗
2,p∗
and divA ∈ Lp (Ω) then g ∈ Lploc (Ω). So by [18, Corollary 9.18] there exists a unique v ∈ Wloc
(Ω)
solution of (2.12). If we write (2.12) as follows
div(A∇v) − div(A)∇v = g,
(2.13)
we have that u and v are solutions, then by [18, Theorem 8.3] we obtain u = vn and the thesis follows.
Instead, if ps = n by [10, Theorem 6.10], A ∈ W 1,q and so u ∈ W 2,q ∩ C 1,1− q (Ω) for all 1 < q < ∞.
Applying the same reasoning to deduce the case of ps < n the thesis follows.
Now, we fix the notation for the coincidence set, non-coincidence set and the free boundary. We
can define:
Λu := {u = 0},
Nu := {u > 0},
Γu = ∂Λu ∩ Ω.
(2.14)
We note that thanks to continuity of u these sets are pointwise defined and we can equivalently write
Γu = ∂Nu ∩ Ω.
3
Weiss’ and Monneau’s quasi-monotonicity formulas
In this section we show that the monotonicity formulas established by Weiss [35] and Monneau [27] in
the Laplace case (A = In ) and by Focardi, Gelli and Spadaro [15] in the A Lipschitz continuous and f
Hölder continuous case, hold in our case as well.
We proceed by fixing the coordinates system: let x0 ∈ Γu , be any point of free boundary, then the
affine change of variables
1
1
x 7−→ x0 + f (x0 )− 2 A 2 (x0 )x = x0 + L(x0 )x
leads to
n
(3.1)
1
E[u, Ω] = f 1− 2 (x0 ) det(A 2 (x0 )) EL(x0 ) [uL(x0 ) , ΩL(x0 ) ],
with the following notations:
ˆ EL(x0 ) [v, A] :=
hCx0 ∇v, ∇vi + 2
A
fL(x0 )
v
f (x0 )
dx
∀ A ⊂ ΩL(x0 ) ,
ΩL(x0 ) := L(x0 )−1 (Ω − x0 ),
(3.2)
uL(x0 ) (x) := u(x0 + L(x0 )x),
fL(x0 ) := f (x0 + L(x0 )x),
1
1
Cx0 := A− 2 (x0 )A(x0 + L(x0 )x)A− 2 (x0 ).
We observe that the image of the free boundary in the new coordinates is:
ΓuL(x0 ) = L(x0 )−1 (Γu − x0 )
5
(3.3)
and we see how energy E is minimized by u if and only if the energy EL(x0 ) is minimized by uL(x0 ) .
Therefore, for a fixed base point x0 ∈ Γu , we change the coordinates system and as we stated before
0 ∈ ΓuL(x0 )
Cx0 (0) = In
fL(x0 ) (0) = f (x0 ).
We identify the two spaces in this section to simplify the ensuing calculations, then with a slight abuse
of notation we reduce to (1.4):
x0 = 0 ∈ Γu ,
A(0) = In ,
f (0) = 1.
We note that with this convention 0 ∈ Ω. In the new coordinates system we define
x
hA(x)ν(x), ν(x)i
if x =
6 0
ν(x) :=
µ(x) :=
per x 6= 0,
1
otherwise.
|x|
(3.4)
We note that µ ∈ C 0 (Ω) by (H1) and (1.4). We prove the following result:
Lemma 3.1. Let A be a matrix-valued field. Assume that (H1), (H2) and (1.4) hold, then
n
µ ∈ W 1,q ∩ C 0,1− p∗ (Ω)
∀q < p∗ ,
(3.5)
and
Λ−1 ≤ µ(x) ≤ Λ
∀x ∈ Ω.
(3.6)
Proof. We prove that µ ∈ W 1,q for any q < p∗ .
We use a characterization of Soblev’s spaces (see [3, proposition IX.3]): µ ∈ W 1,q (Ω)if and only if
exists a constant C > 0 such that for every open ω ⊂⊂ Ω and for any h ∈ Rn with |h| < dist(ω, ∂Ω) it
holds
kτh µ − µkLq (ω) ≤ C |h|.
For the convexity of the function | · |q , remembering that A is (1 −
|x|
− x ≤ 2|h|, we have
all x ∈ Rn and h 6= −x holds (x + h) |x+h|
n
p∗ )-Hölder
continuous and since for
q
ˆ x+h x+h
x x =
hA(x + h) |x + h| , |x + h| i − hA(x) |x| , |x| i dx
ω
q
ˆ
x+h x+h
x+h
x
x+h
x
=
h(A(x + h) − A(x)) |x + h| , |x + h| i + h(A(x) − A(0)) |x + h| + |x| , |x + h| − |x| i dx
ω
q q ˆ
ˆ
|x|
q−1
q
q A(x) − A(0) ≤2
|A(x + h) − A(x)| dx +
2 (x + h)
− x
|x|
|x + h|
ω
ω
ˆ
1
≤ 2q−1 c|h|q + 4q |h|q
dx = C |h|q ,
n q∗
ω |x| p
kτh µ−µkqLq (ω)
nq
where in the last equality, we rely on n|x|− p∗ being integrable if and only if q < p∗ . By Sobolev
immersion theorem, we have µ ∈ C 0,1− q for any q < p∗ .
Thanks to the structure of µ we can earn more regularity. In particular µ ∈ C 0,γ with γ = 1 − pn∗ .
We start off proving the inequality when one of the two points is 0:
x x
x x
x x |µ(x) − µ(0)| = hA(x) ,
i − 1 = hA(x) ,
i−h ,
i
|x| |x|
|x| |x|
|x| |x| x x = h(A(x) − A(0)) ,
i = [A]C 0,γ |x|γ .
|x| |x| x
Assume now that x, y 6= 0 and prove the inequality in the remaining case. Let z = |y| |x|
then
|µ(x) − µ(y)| ≤ |µ(x) − µ(z)| + |µ(z) − µ(y)|.
As
z
|z|
=
x
|x|
x x |µ(x) − µ(z)| = h(A(x) − A(z)) ,
i ≤ [A]C 0,γ |x − z|γ ,
|x| |x|
6
while by |z| = |y| = r
z z
z z z z
y y y y |µ(z) − µ(y)| = hA(z) , i − hA(y) , i ≤ h(A(z) − A(y)) , i + hA(y)) , i − hA(y) , i
r r
r r
r r
r r
r r
1−γ
z+y z−y |z − y|
γ
γ
γ
≤ [A]C 0,γ |z − y| + h(A(y) − A(0))
|z − y|
,
i ≤ [A]C 0,γ |z − y| + 2
r
r r1−γ
≤ [A]C 0,γ |z − y|γ + 21−γ |z − y|γ ≤ C[A]C 0,γ |z − y|γ .
Therefore, since |x − z| = ||x| − |y|| ≤ |x − y| and |z − y| ≤ |z − x| + |x − y| ≤ 2|x − y| we have the thesis
|µ(x) − µ(y)| ≤ C[A]C 0,γ |x − y|γ .
To study the regularity of the free boundary, we analyze the functions of the rescaled functions
ur (x) =
u(rx)
.
r2
(3.7)
∗
The rescaled function satisfies an appropriate PDE and satisfies uniform W 2,p estimates thanks to
the regularity theory for elliptic equations.
Proposition 3.2. Let u be a solution to the obstacle problem (2.7) and assume (1.4) holds. Then,
for every R > 0 there exists a constant C > 0 such that, for every r ∈ (0, dist(0,∂Ω)
)
4R
kur kW 2,p∗ (BR ) ≤ C.
(3.8)
0
In particular, the functions ur are equibounded in C 1,γ for γ 0 ≤ γ := 1 −
n
p∗ .
Proof. By (3.7) and proposition 2.7 holds
div(A(rx)∇ur (x)) = f (rx)χ{ur >0} (x) a.e. on B4R and on D0 (B4R ),
(3.9)
∗
and ur ∈ W 2,p ∩ C 1,γ (B4R ). By (1.4) we have 0 ∈ Γu , then ur (0) = 0. Since ur ≥ 0, by theorems
8.17 and 8.18 of [18] we have
kur kL∞ (B4R ) ≤ C(R)kf kL∞ (B4R ) .
(3.10)
Thanks to theorem 8.32 [18] and (3.10) we obtain
kur kC 1,γ (B2R ) ≤ C kur kL∞ (B4R ) + kf kL∞ (B4R ) ≤ C 0 kf kL∞ (B4R ) .
(3.11)
We observe that, as in proposition 2.7, ur is solution to
A(rx) : ∇2 ur (x) = f (rx)χ{ur >0} − rhdivA(rx), ∇ur (x)i =: gr (x),
(3.12)
∗
with gr ∈ Lp (B2R ), then by theorem 9.11 [18]
kur kW 2,p∗ (BR ) ≤ C kur kLp∗ (B2R ) + kgkLp∗ (B2R ) .
Then by (3.11)
∗
kgr kpLp∗ (B2R )
ˆ
(3.13)
∗
|f (rx)χ{ur >0} − rhdivA(rx), ∇ur (x)i|p dx
B2R
ˆ
p∗
p∗ −n
p∗
≤ C kf kL∞ (B2R ) 1 + r
|hdivA(y)| dy
B2rR
dist (0, ∂Ω) p∗ −n
p∗
p∗
≤ C kf kL∞ (B2R ) 1 +
kdivA(y)kW 1,p∗ (Ω) .
4R
=
So kur kW 2,p∗ (BR ) ≤ C, where C does not depend on r.
Remark 3.3. Recalling (1.4) we have u(0) = 0 and ∇u(0) = 0 so
kukL∞ (Br ) ≤ C r2
and
k∇ukL∞ (Br ) ≤ C r.
(3.14)
We note that the constant in (3.14) depends only on the constant C in (3.8) therefore it is uniformly
bounded for points x ∈ Γu ∩ K for each compact set K ⊂ Ω.
7
We introduce rescaled volume and boundary energies
ˆ
E(r) := E[u, Br ] =
(hA(x)∇u(x), ∇u(x)i + 2f (x)u(x)) dx
Br
ˆ
= rn+2
(hA(rx)∇ur (x), ∇ur (x)i + 2f (rx)ur (x)) dx
(3.15)
B1
ˆ
ˆ
µ(x)u2 (x) dH n−1 = rn+3
H(r) :=
∂Br
µ(rx)u2r (x) dH n−1 .
(3.16)
∂B1
We now introduce an energy “à la Weiss” combining and rescaling the above terms:
Φ(r) := r−n−2 E(r) − 2 r−n−3 H(r).
Remark 3.4. By (1.4), (3.15), (3.16) and Proposition 3.2 we have
ˆ
(3.14)
E(r) =
(|∇ur |2 + 2u) dx + O(rn+2+min(γ,α) ) = O(rn+2 ),
B
ˆ r
(3.14)
H(r) =
u2 dH n−1 + O(rn+3+γ ) = O(rn+3 ).
(3.17)
(3.18)
∂Br
Hence, the choice of the renormalized factors in (3.17).
3.1
Derivatives of E and H
We provide some estimates for the derivatives of E and H. To this aim, following Focardi, Gelli and
Spadaro [15], we need to use a generalization of Rellich–Necas’ identity due to Payne–Weinberger [15,
Lemma 3.4] in order to calculate the derivative of E(r). We refer to [15, Proposition 3.5] for details of
∗
proof, where to estimate E 0 (r) we recall that A ∈ W 1,p .
Proposition 3.5. There exists a constant C1 > 0, C1 = C1 (λ, C, kAkW 1+s,p (Ω) ), such that for L1 -a.e.
r ∈ (0, dist(0, ∂Ω)),
ˆ
ˆ
1
0
−1
2
n−1
E (r) = 2
µ hAν, ∇ui dH
+
hA∇u, ∇ui div(µ−1 Ax) dx
r Br
∂Br
ˆ
ˆ
2
2
−1
−
f hµ Ax, ∇ui dx −
hA∇u, ∇T (µ−1 Ax)∇ui dx
r Br
r Br
ˆ
+2
f u dH n−1 + ε(r),
∂Br
with ε(r) ≤ C1 E(r) r
− pn∗
.
The next step is to estimate the derivative of H(r). By definition H(r) is a boundary integral;
our strategy will be to bring us back to a volume integral using the divergence theorem and deriving
through coarea’s formula. In this way we will find, integrated on ∂Br , the function divA that by (H1)
is a function in W s,p (Ω) with s > p1 that a priori is not well defined on ∂Br . Then, taking into account
the concept of trace we prove a corollary of the Coarea formula.
Proposition 3.6. Let g ∈ W s,p (B1 ) with s > p1 . Then for L1 -a.e. r ∈ (0, 1) it holds
ˆ
ˆ
d
g dx =
γ0 (g) dH n−1 .
dr
Br
∂Br
(3.19)
W s,p (B1 )
Proof. Let (gj )j ⊂ C ∞ (B1 ) such that gj −−−−−−→ g. For each function gj , by the coarea formula for
L1 -a.e. r ∈ (0, 1) it holds
ˆ
ˆ
d
gj dx =
gj dH n−1 ,
(3.20)
dr
Br
∂Br
By continuity of trace and dominated convergence theorem we have
ˆ
ˆ
lim
gj dH n−1 =
γ0 (g) dH n−1 .
j
Let us now prove
d
limj dr
´
Br
∂Br
gj dx
=
∂Br
d
dr
´
g dx .
Br
8
(3.21)
To this aim we define the function G(r) :=
´
Br
g dx and the sequence Gj (r) :=
´
Br
gj dx; we prove
W 1,1 (0,1)
that Gj −−−−−−→ G.
We recall the characterization of functions W 1,1 on a interval as absolutely continuous functions.
In order to deduce that G, Gj ∈ W 1,1 we have to prove that for any ε > P
0 there exists an δ > 0 such
that
for
any
finite
sequence
of
disjoint
intervals
(a
,
b
)
of
(0,
1)
holds
k
k
k |Gj (bk ) − Gj (ak )| < ε if
P
|b
−
a
|
<
δ.
k
k
k
Therefore, we estimate as follows
ˆ
X
Xˆ
|Gj (bk ) − Gj (ak )| =
gj dx ≤ S
|gj | dx < ε,
k
Bbk \Bak
k
k (Bbk \Bak )
where in the last inequality, we used the absolute continuity of the integral and
ˆ
ˆ 1
n
n−1
L (∪k (Bbk \ Bak ) = nωn
r
dr ≤ nωn
rn−1 dr ≤ nωn δ.
∪k (bk ,ak )
1−δ
The previous argument holds for G as well. Thus G and Gj are differentiable L1 -a.e. (0, 1). On the
other hand by the Coarea formula, we can represent the weak derivative of Gj in the following way:
ˆ
ˆ
G0j (r) =
gj dH n−1 , G0 (r) =
g dH n−1
L1 -a.e. r ∈ (0, 1).
∂Br
Thus
∂Br
ˆ
1
kGj − GkL1 ((0,1)) =
0
ˆ
kG0j − G0 kL1 ((0,1)) =
0
ˆ
Br
1
ˆ
j→∞
(gj − g) dx dr ≤ kgj − gkL1 (B1 ) −−−→ 0,
∂Br
j→∞
(gj − g) dH n−1 dr ≤ kgj − gkL1 (B1 ) −−−→ 0,
therefore up to subsequence G0j → G0 L1 -a.e. (0, 1); then combining together this, (3.20) and (3.21)
we have the thesis.
We estimate the derivative of H(r).
n−sp
,s ;
Before proceeding we introduce a notation. We fix s, p and as in Remark 2.5 choose t0 ∈ p(n−1)
np
n−sp
n
n
morever we ask that p > 1+s−t
(in
this
way
So
we
choose
t
∈
.
>
n).
,
s
∧
(s
+
1
−
)
0
n−(s−t0 )p
p(n−1)
p
0
We observe that there exists such value t0 if and only if p >
p
Θ = Θ(s, p, n, t0 ) =
np
n−(s−t0 )p
n2
n(1+s)−1
∧ n. Then we define:
if p > n
if p ≤ n.
(3.22)
n
Remark 3.7. We observe that by definition of (3.22) the function r 7→ r− Θ(s) ∈ L1 ((0, 1)) if and only
n
n2
if p > n(1+s)−1
∧ n. The integrability of r− Θ(s) is the motivation for the choice of condition (H1).
Proposition 3.8. There exists a positive constant C2 = C2 (kAkW 1+s,p ) such that for L1 -a.e. r ∈
(0, dist(0, ∂Ω)) it holds
ˆ
n−1
H0 (r) =
H(r) + 2
uhAν, ∇ui dH n−1 + h(r),
(3.23)
r
∂Br
n
with |h(r)| ≤ C2 H(r)r− Θ , Θ defined in (3.22).
Proof. By the divergence theorem we write H(r) as volume integral
ˆ
ˆ
1
1
H(r) =
u2 (x)hA(x)x, νi dH n−1 =
div u2 (x)A(x)x dx
r ∂Br
r Br
ˆ
ˆ
ˆ
2
1
1
=
u∇u · A(x)x dx +
u2 (x)TrA dx +
u2 (x)divA(x) · x dx.
r Br
r Br
r Br
By taking coarea formula and Proposition 3.6 into account, we have
ˆ
ˆ
ˆ
1
1
1
H0 (r) = − H(r) + 2
uhAν, ∇ui dH n−1 +
u2 TrA dH n−1 +
u2 γ0 divA(x) · x dH n−1
r
r ∂Br
r ∂Br
∂Br
ˆ
n−1
=
H(r) + 2
µhAν, ∇ui dH n−1 + h(r),
r
∂Br
9
ˆ
ˆ
1
1
2
n−1
h(r) =
u TrA − nµ dH
+
u2 γ0 divA(x) · x dH n−1 =: I + II.
r ∂Br
r ∂Br
We estimate separately the two terms.
For the first term let’s recall the Hölder continuity of A and µ, (3.14), that by (1.4) A(0) = In and
by the definition µ(0) = 1:
ˆ
ˆ
X
X
1
1
u2
|I| = aii (x) − µ(x) dH n−1 ≤
u2
|aii (x) − aii (0)| + |µ(0) − µ(x)| dH n−1
r
r
with
∂Br
0
≤Cr
∂Br
i
n+3− pn∗
0
≤ C H(r) r
− pn∗
i
,
(3.24)
where in the last inequality, we used (3.18).
For the second term by Hölder inequality, by (3.14) and recalling by Remark 2.5 that γ0 (divA) ∈
L1 (∂Br ) we have:
ˆ
1
|II| ≤
u2 γ0 divA (x) |x| dH n−1 ≤ C 0 r4 kγ0 (divA)kL1 (∂Br ) .
(3.25)
r ∂Br
Now we analyze separately the two cases p > n and p ≤ n.
We start off with the case p > n. We use (2.4), (3.18) in (3.25) to get
n
n
n
|II| ≤ C kdivAkW s,p (Br ) rn+3 r− p ≤ C kdivAkW p,s (Ω) H(r) r− p ≤ C H(r) r− p .
(3.26)
If p ≥ n by (2.5) we have
kγ0 (divA)kL1 (∂Br ;H n−1 ) ≤ C rn−1 r−
n−(s−t0 )p
p
kdivAk
W
t0 ,
np
n−(s−t0 )p
(Br )
.
Hence, recalling (3.25) and (3.18)
|II| ≤ C kdivAk
W
t0 ,
np
n−(s−t0 )p
(Br )
≤ C kdivAkW s,p (Ω) H(r) r
−
rn+3 r−
n−(s−t0 )p
p
n−(s−t0 )p
p
≤ C kdivAk
np
t0 ,
n−(s−t0 )p
W
≤ C H(r) r
−
(Ω)
H(r) r−
n−(s−t0 )p
p
n−(s−t0 )p
p
(3.27)
So, assuming the notation introduced in (3.22), combining together (3.24), (3.27) and (3.26), and
recalling that Θ < p∗ , we have
n
n
n
|h(r)| ≤ C 0 H(r) r− p∗ + C̄H(r) r− Θ ≤ C2 H(r) r− Θ .
3.2
Weiss’s quasi-monotonicity formula
In this section we prove a Weiss’ quasi-monotonicity formula that implies the 2-homogeneity of blow-ups
of u in free boundary points.
Theorem 3.9. Assume that (H1)-(H3) and (1.4) hold. There exist nonnegative constants C¯3 and C4
independent on r such that the function
ˆ r
n
1− n
n
ω(t) C¯3 t1− Θ(s)
− Θ(s)
C¯3 r Θ(s)
r 7→ Φ(r) e
+
e
+ C4
t
dt
t
0
is nondecreasing on (0, 21 dist(0, ∂Ω) ∧ 1).
More precisely, the following estimate holds true for L1 -a.e. r in such an interval:
ˆ r
n
1− n
n
d
ω(t) C¯3 t1− Θ(s)
− Θ(s)
C¯3 r Θ(s)
Φ(r) e
+ C4
t
+
e
dt
dr
t
0
n
¯ 1− Θ(s) ˆ
2eC3 r
u 2
−1
dH n−1 .
≥
µ
hµ
Aν,
∇ui
−
2
rn+2
r
∂Br
(3.28)
In particular, the limit Φ(0+ ) := limr→0+ Φ(r) exists and it is finite and there exists a constant c > 0
such that
Φ(r) − Φ(0+ )
¯
≥ Φ(r) eC3 r
1−
n
Θ(s)
ˆ
+ C4
0
r
n
t− Θ(s) +
ω(t)
t
¯
1−
eC3 t
10
n
Θ(s)
(3.29)
n
dt − Φ(0+ ) − c r1− Θ(s) + ω(r) .
Proof. Set s >
(3.17):
1
p
we will indicate for convenience Θ(s) only with Θ. Assume the definition of Φ(r) by
Φ(r) := r−n−2 E(r) − 2 r−n−3 H(r).
Then for L1 -a.e. r ∈ dist(0, ∂Ω) we have
Φ0 (r) =
E(r)
H0 (r)
H(r)
E 0 (r)
− (n + 2) n+3 − 2 n+3 + 2(n + 3) n+4 .
n+2
r
r
r
r
(3.30)
By Proposition 3.5 we have
ˆ
ˆ
E 0 (r)
E(r)
2
1
−1
2
n−1
− (n + 2) n+3 ≥ n+2
µ hAν, ∇ui dH
hA∇u, ∇ui div(µ−1 Ax) dx
+ n+3
rn+2
r
r
r
∂Br
Br
ˆ
ˆ
ˆ
2
2
2
−1
T
−1
f hµ Ax, ∇ui dx − n+3
hA∇u, ∇ (µ Ax)∇ui dx + n+2
f u dH n−1
− n+3
r
r
r
Br
Br
∂Br
ˆ
ˆ
2(n + 2)
C1 E(r) n + 2
hA∇u, ∇ui dx −
f u dx.
− n+2 n∗ − n+3
r
r
rn+3
rp
Br
Br
Then, integrating by parts and given (2.11):
ˆ
ˆ
ˆ
ˆ
hA∇u, ∇ui dx +
f u dx =
hA∇u, ∇ui dx +
Br
Br
Br
ˆ
uhAν, ∇ui dH n−1 .
udiv(A∇u) dx =
Br
∂Br
(3.31)
Thus, using (3.31) on 4 times, we deduce
ˆ
E(r)
C1
2
E 0 (r)
− pn∗
−
(n
+
2)
≥
−
E(r)
r
+
µ−1 hAν, ∇ui2 dH n−1
rn+2
rn+3
rn+2
rn+2 ∂Br
ˆ 1
+ n+3
hA∇u, ∇ui div(µ−1 Ax) − 2hA∇u, ∇T (µ−1 Ax)∇ui − (n − 2)hA∇u, ∇ui dx
r
B
ˆ r
ˆ
ˆ
ˆ
2
4
2n
2
f hµ−1 Ax, ∇ui dx + n+2
f u dH n−1 − n+3
uhAν, ∇ui dH n−1 − n+3
f u dx.
− n+3
r
r
r
r
Br
∂Br
∂Br
Br
(3.32)
Instead the proposition 3.8 leads to
−2
n
H0 (r)
H(r)
2C2
4
8
+ 2(n + 3) n+4 ≥ − n+3 H(r)r− Θ + n+4 H(r) − n+3
rn+3
r
r
r
r
ˆ
uhAν, ∇ui dH n−1 .
Combining together (3.32) and (3.33) and since p∗ ≥ Θ we finally infer that
ˆ
n
2
Φ0 (r) + (C10 ∨ C2 )Φ(r)r− Θ ≥ n+2
µ−1 hAν, ∇ui2 dH n−1
r
∂Br
ˆ 1
−1
hA∇u, ∇ui div(µ Ax) − 2hA∇u, ∇T (µ−1 Ax)∇ui − (n − 2)hA∇u, ∇ui dx
+ n+3
r
ˆBr
ˆ
ˆ
2
2
4
− n+3
f hµ−1 Ax, ∇ui dx + n+2
f u dH n−1 − n+3
uhAν, ∇ui dH n−1
r
r
r
∂Br
∂Br
ˆBr
ˆ
2n
8
4
n−1
− n+3
f u dx + n+4 H(r) − n+3
uhAν, ∇ui dH
r
r
r
Br
∂Br
ˆ
u
2
u2
= n+2
µ−1 hAν, ∇ui2 + 4 2 µ − 4 hAν, ∇ui dH n−1
r
r
r
r
ˆ ∂B
1
+ n+3
hA∇u, ∇ui div(µ−1 Ax) − 2hA∇u, ∇T (µ−1 Ax)∇ui − (n − 2)hA∇u, ∇ui dx
r
B
ˆr
ˆ
2
− n+3
f hµ−1 Ax, ∇ui − nu dx − r
f u dx =: R1 + R2 + R3 .
r
Br
∂Br
We estimate separately three addends.
ˆ
2
u2
u
R1 = n+2
µ µ−2 hAν, ∇ui2 + 4 2 − 4µ−1 hAν, ∇ui dH n−1
r
r
r
∂B
ˆ r 2
2
u
= n+2
µ hµ−1 Aν, ∇ui − 2
dH n−1 .
r
r
∂Br
11
(3.33)
∂Br
(3.34)
(3.35)
Since n = div(x) by (3.14) we have
ˆ
1 −1
T
−1
hA∇u, ∇ui div(µ Ax) − 2hA∇u, ∇ (µ Ax)∇ui − (n − 2)hA∇u, ∇ui dx
|R2 | = n+3 r
Br
ˆ 1
= n+3
hA∇u, ∇ui div(µ−1 Ax − x) − 2hA∇u, ∇T (µ−1 Ax − x)∇ui dx
r
Br
ˆ ˆ
2
C Λ
C0 Λ ≤ n+1
|div(µ−1 Ax − x) + 2|∇(µ−1 Ax − x)| dx ≤ n+1
|∇(µ−1 Ax − x)| dx,
r
r
Br
Br
We can only estimate |∇(µ−1 Ax − x)|:
|∇(µ−1 Ax − x)| = ∇ (µ−1 A − id)x = |∇(µ−1 A − id)x + (µ−1 A − id)|
n
= |∇(µ−1 ) ⊗ Ax + µ−1 ∇A x + (µ−1 A − In )| ≤ Λr(|∇A| + |∇µ|) + C r1− p∗ ,
where in the last inequality, we have used the γ-Holderian continuity of A − µIn . Thus, by Lemma 3.1
ˆ ˆ n
C 0Λ
C 00
|R2 | ≤ n+1
|∇(µ−1 Ax − x)| dx ≤ n+1
r1− p∗ + r(|∇A| + |∇µ|) dx
r
r
Br
Br
ˆ
1
ˆ
q1
00
n
1
∗
1
n
C
p∗
(ωn rn )1− p∗ +
≤ C 000 r− p∗ + n
(|∇A|p ) dx
(|∇µ|q ) dx (ωn rn )1− q ≤ C̄r− q ,
r
Br
Br
for each n < Θ < q < p∗ , whence
|R2 | ≤ c
E(r) − nq
r .
rn+2
Moreover, by (3.16) and (3.6)
H(r)
≤ ckur k2L∞ ≤ c,
rn+3
by a certain constant c independent from r, then
n
n
n
n
E(r)
H(r)
H(r)
|R2 | ≤ c
−
2
r− q + 2c n+3 r− q ≤ c Φ(r)r− Θ + c r− Θ .
n+2
n+3
r
r
r
0≤
(3.36)
Finally, assuming that n = divx and using the following identity, consequence of the divergence theorem
ˆ
ˆ
(hx, ∇ui + udivx) dx = r
u dH n−1 ,
(3.37)
Br
∂Br
we have
R3 = −
2
ˆ
ˆ
−1
Ax, ∇ui − nu dx − r
n−1
f hµ
f u dH
rn+3
B
∂Br
ˆ r
ˆ
2
= − n+3
(f (x) − f (0) hµ−1 Ax, ∇ui + f (0)
hµ−1 Ax, ∇ui − udivx dx
r
Br
Br
ˆ
ˆ
−r
(f (x) − f (0))u dx − r f (0)
u dx
∂Br
∂Br
ˆ
ˆ
2
(f (x) − f (0) hµ−1 Ax, ∇ui + f (0)
= − n+3
hµ−1 Ax − x, ∇ui dx
r
Br
Br
ˆ
−r
(f (x) − f (0))u dH n−1 .
∂Br
Thus
ˆ
2 f
(0)
hµ−1 Ax − x, ∇ui dx
n+3
r
Br
ˆ
ˆ
+
(f (x) − f (0) hµ−1 Ax, ∇ui − r
(f (x) − f (0))u dH n−1 Br
∂Br
ˆ
ˆ
ˆ
c
n−1
≤ n+1
|A − µIn | dx +
|f (x) − f (0)| dx + r
|f (x) − f (0)| dH
r
Br
Br
∂Br
n
n
c
ω(r)
≤ n+1 (rn+1− p∗ + rn ω(r)) ≤ c r− p∗ +
.
r
r
|R3 | =
12
(3.38)
Now combining together (3.34), (3.35), (3.36) e (3.38)we have
ˆ
n
n
u 2
ω(r)
2
0
−Θ
−Θ
µ hµ−1 Aν, ∇ui − 2
Φ (r) + C3 Φ(r) r
+ C4 r
+
≥ n+2
dH n−1 .
r
r
r
∂Br
¯
Multiplying the inequality by the integral factor eC3 r
n
Φ(r) e
C¯3 r 1− Θ
0
1− n
Θ
with C¯3 =
C3
n
1− Θ
(3.39)
we get
n ˆ
¯ 1−
n
u 2
ω(r) C¯3 r1− Θn
2eC3 r Θ
−1
−Θ
µ
hµ
Aν,
∇ui
−
2
+ C4 r
+
e
dH n−1
≥
r
rn+2
r
∂Br
whence
n ˆ
ˆ r
¯ 1−
n
n
d
ω(t) C¯3 t1− Θn
2eC3 r Θ
u 2
C¯3 r 1− Θ
−Θ
Φ(r) e
+ C4
t
+
e
dt ≥
µ hµ−1 Aν, ∇ui − 2
dH n−1 .
n+2
dr
t
r
r
0
∂Br
(3.40)
In particular, the quantity under the sign of the derivative, bounded by construction, is also monotonic,
therefore its limit exists when r → 0+ . It follows that Φ(0+ ) := limr→0+ Φ(r) exists and is bounded.
Finally
ˆ r
n
n
n
ω(t) C¯3 t1− Θn
+
C¯3 r 1− Θ
C¯3 r 1− Θ
−Θ
+
Φ(r) − Φ(0 ) ≥ − |Φ(r) e
− Φ(r)| + Φ(r) e
+ C4
t
dt
e
t
0
ˆ r
n
ω(t) C¯3 t1− Θn
− Φ(0+ ) − C4
e
dt
t− Θ +
t
0
ˆ
r
n
n
n
ω(t) C¯3 t1− Θn
0 1− Θ
C¯3 r 1− Θ
−Θ
e
≥ − |Φ(r)| c r
+ C4
dt
+ Φ(r) e
t
+
t
0
n
− Φ(0+ ) − c0 r1− Θ + ω(r)
ˆ r
n
n
n
ω(t) C¯3 t1− Θn
−Θ
C¯3 r 1− Θ
+ C4
t
+
e
dt − Φ(0+ ) − c r1− Θ + ω(r) ,
≥Φ(r) e
t
0
(3.41)
where in the last inequality, we used the boundedness of Φ(r).
3.3
Monneau’s quasi-monotonicity formula
In this section we prove a Monneau’s type quasi-monotonicity formula (see [27]) for singular free
boundary points.
Let v be a 2-homogeneous positive polynomial, solving
on Rn .
∆v = 1
Let
Ψv (r) :=
1
rn+2
ˆ
|∇v|2 + 2v dx −
Br
2
rn+3
(3.42)
ˆ
v 2 dH n−1 .
(3.43)
∂Br
We note that the expression of Ψv (r) is analogous to those of Φ with coefficients frozen in 0 (recalling
(3.17)). An integration by parts, (3.43) and the 2-homogeneity of v yield
ˆ
ˆ
ˆ
ˆ
1
1
1
1
2
n−1
|∇v| dx = n+2
div(v∇v) − v ∆v dx = n+3
h∇v, xi dH
− n+2
v dx
rn+2 Br
r
r
r
Br
ˆBr
ˆ
ˆ ∂Br
ˆ
1
1
= n+3
v 2 dH n−1 − n+2
v dx =
v 2 dH n−1 −
v dx
r
r
∂Br
Br
∂B1
B1
ˆ
and therefore
Ψv (r) = Ψv (1) =
v dx.
(3.44)
B1
In the next theorem we give a monotonicity formula for solutions of the obstacle problem such that 0
is a point of the free boundary and
Φ(0+ ) = Ψv (1) for some v 2-homogeneous solution of (3.42).
(3.45)
As it will be explained in Definition 4.8 formula (3.45) characterizes the singular part of the free
boundary.
13
Theorem 3.10. Assume (H1)-(H3) and (1.4). Let u be the minimizer of E on K, with 0 ∈ Sing(u)
(i.e. (3.45) holds), and v be as above. Then, there exists a positive constant C5 = C5 (λ, kAkW s,p ) such
that
ˆ
n
(3.46)
r 7−→
(ur − v)2 dH n−1 + C5 r(1− Θ(s) ) + ω(r)
∂B1
1
2 dist(0, ∂Ω)
is nondecreasing on (0,
∧ 1). More precisely, L1 -a.e. on such an interval
ˆ
ˆ r
n
ω(t)
d
(ur − v)2 dH n−1 + C5 r1− Θ(s) +
dt
dr
t
0
∂B1
(3.47)
ˆ r
n
1−
1− n
n
ω(t)
2 C3 r Θ(s)
Θ(s)
≥
e
Φ(r) + C4
ec 3 t
t− Θ(s) +
dt − Ψv (1) .
r
t
0
Proof. Set wr = ur − v. As v is 2-homogenus we have that wr (x) = w(rx)
r 2 . Assuming that by (1.4)
A(0) = In , by the divergence theorem and Euler’s theorem we find
ˆ
ˆ
d
d w(rx)
wr2 dH n−1 =
wr ( 2 ) dH n−1
dr ∂B1
dr r
∂B1
ˆ
ˆ
2
2
=
wr (h∇wr , xi − 2wr ) dH n−1 =
wr (h∇ur , xi − 2ur ) dH n−1
r ∂B1
r ∂B1
ˆ
ˆ
2
2
n−1
=
wr (hA(rx)∇ur , xi − 2ur ) dH
+
wr h(A(0 − A(rx)))∇ur , xi dH n−1
r ∂B1
r ∂B1
ˆ
n
2
≥
wr (hA(rx)∇ur , xi − 2ur ) dH n−1 − C k∇ur kL2 (∂B1 ) kwr kL2 (∂B1 ) [A]0,γ r− p∗ ,
r ∂B1
thus by (3.14)
d
dr
ˆ
wr2
∂B1
dH
n−1
2
≥
r
ˆ
n
wr (hA(rx)∇ur , xi − 2ur ) dH n−1 − C r− p∗ .
(3.48)
∂B1
Using an integration by parts, and (3.42) we can rewrite the first term of the right as
ˆ
wr (hA(rx)∇ur , xi − 2ur ) dH n−1
∂B1
ˆ
ˆ
(2.11)
=
hA(rx)∇ur , ∇wr i + wr f (rx)χ{ur >0} (x) dx −
2 wr ur dH n−1
B
∂B
ˆ 1
ˆ 1
hA(rx)∇ur , ∇ur i + ur f (rx)χ{ur >0} (x) dx −
2 u2r dH n−1
=
B1
ˆ
ˆ∂B1
−
hA(rx)∇ur , ∇vi + v f (rx)χ{ur >0} (x) dx +
2 v ur dH n−1
B1
∂B1
ˆ
ˆ
=Φ(r) −
f (rx) ur + v χ{ur >0} (x) dx + 2
µ(rx) − µ(0) u2r dH n−1
B1
∂B1
ˆ
ˆ
−
hA(rx)∇ur , ∇vi dx + 2
v ur dH n−1
B1
∂B1
ˆ
ˆ
ˆ
≥Φ(r) +
(ur + v) dx −
h∇ur , ∇vi dx −
f (rx) − f (0) (ur + v) dx
B1
B1
B1
ˆ
ˆ
ˆ
2
n−1
−
h A(rx) − A(0) ∇ur , ∇vi dx + 2
µ(rx) − µ(0) ur dH
+2
v ur dH n−1 .
B1
∂B1
∂B1
(3.49)
14
Recalling the γ-Hölderianity of A and µ, by divergence theorem, we obtain
ˆ
wr (hA(rx)∇ur , xi − 2ur ) dH n−1
∂B1
ˆ
ˆ
ˆ
≥Φ(r) +
(ur + v) dx −
h∇ur , ∇vi dx + 2
v ur dH n−1 − c (rγ + ω(r))
B1
B1
∂B1
ˆ
ˆ
ˆ
(3.44)
= Φ(r) − Ψv (1) +
(ur ∆v) dx −
h∇ur , ∇vi dx + 2
v ur dH n−1 − c0 (rγ + ω(r))
B1
B1
∂B1
ˆ
ˆ
=Φ(r) − Ψv (1) +
(div(ur ∇v) dx + 2
v ur dH n−1 − c0 (rγ + ω(r))
∂B1
ˆB1
=Φ(r) − Ψv (1) +
ur (h∇v, xi dx + 2v dH n−1 − c0 (rγ + ω(r)) = Φ(r) − Ψv (1) − c0 (rγ + ω(r)) .
∂B1
(3.50)
So, combining together (3.48) and (3.50), and assuming that γ := 1 −
d
dr
ˆ
wr2 dH n−1 ≥
∂B1
2
Φ(r) − Ψv (1) − c0
r
n
n
p∗
r − p∗ +
we deduce
ω(r)
r
.
by inequality (3.29) we deduce
ˆ
ˆ r
n
n
n
ω(t) C¯3 t1− Θ(s)
d
2
¯ 1− Θ(s)
wr2 dH n−1 ≥ Φ(r)eC3 r
+ C4
dt
t− Θ(s) +
e
dr ∂B1
r
t
0
n
ω(r)
1− Θ(s)
− pn∗
0
+ ω(r) − Ψv (1) − c r
−c r
+
r
ˆ
r
n
n
1−
1−
n
ω(t) C¯3 t Θ(s)
2
Θ(s)
¯
+ C4
dt − Ψv (1)
≥ Φ(r)eC3 r
t− Θ(s) +
e
r
t
0
n
ω(r)
− c00 r− Θ(s) +
r
and then set C5 =
d
dr
ˆ
c00
n
1− Θ(s)
wr2 dH n−1 + C5
n
ˆ
r1− Θ(s) +
∂B1
≥
4
n
2
¯ 1− Θ(s)
Φ(r)eC3 r
r
r
ω(t)
dt
t
0
ˆ r
n
n
ω(t) C¯3 t1− Θ(s)
e
+ C4
dt − Ψv (1) .
t− Θ(s) +
t
0
Regularity of the free boundary
Using the quasi-monotonicity formulas in Section 3, we study the regularity of the free boundary for
the obstacle problem (2.7) following the nergetic approach in [15].
4.1
Blow-ups
In this paragraph we shall investigate the existence and uniqueness of the blow-ups. To this aim,
we need to introduce a new notation for the rescaled functions in any free boundary point x0 ∈ Γu
similarly to (3.7)
u(x0 + rx)
ux0 ,r :=
.
(4.1)
r2
Remark 4.1. We note that by Proposition 3.2 it follows the uniform boundedness of the sequence
(ux0 ,r )r in C 0,γ (Rn ). Moreover, for base points x0 in a compact set of Ω, the C 0,γ norms and thus,
the constants in the monotonicity formulas are uniformly bounded. Indeed, as pointed out in the
corresponding statements they depend on kAkW s,p (Ω) and dist(x0 , ∂Ω).
15
We recall the notation introduced in Section 3
ΩL(x0 ) := L(x0 )−1 (Ω − x0 ),
uL(x0 ) (x) := u(x0 + L(x0 )y),
fL(x0 ) := f (x0 + L(x0 )y),
1
1
Cx0 := A− 2 (x0 )A(x0 + L(x0 )y)A− 2 (x0 ),
and in addition as in (3.4) and (3.17) we set
u(x0 + rL(x0 )y)
,
r2
µL(x0 ) (y) :=hCx0 (y)ν(y), ν(y)i
y 6= 0,
µL(x0 ) (0) := 1,
ˆ
fL(x0 ) (ry)
ΦL(x0 ) (r) :=
hCx0 (ry)∇uL(x0 ),r (y), ∇uL(x0 ),r (y)i + 2
uL(x0 ),r dy
f (x0 )
B1
ˆ
−2
µL(x0 ) (ry) u2L(x0 ),r (y) dH n−1
uL(x0 ),r (y) :=
(4.2)
∂B1
Remark 4.2. We note by the definition that Λ−2 ≤ µL(x0 ) (y) ≤ Λ2 and µL(x0 ) ∈ C 0,γ (Ω).
As in classical case, the solution u presents a quadratic growth. The lacking regularity of the
problem does not allow us to use the classic approach of Caffarelli [6] used also by Focardi, Gelli and
Spadaro in [15, Lemma 4.3]. We use the approach of Blank and Hao in [2, Chapter 3].
Proposition 4.3 ( [2, Theorem 3.9]). Let x0 ∈ Γu , and u be minimum of (2.7). Then, there exists a
constant θ > 0 such that
sup u ≥ θ r2 .
(4.3)
∂Br (x0 )
Remark 4.4. From quadratic growth we obtain that, if v(y) = w(L−1 (x0 )y) is a limit of C 1,γ a
converging sequence of rescalings (ux0 ,rj )j in a free boundary point x0 ∈ Γu , then 0 ∈ Γw , i.e. w 6≡ 0
in any neighborhood 0.
It’s possible to give a classification of blow-ups. We begin by recalling the result in the classical
case established by Caffarelli [4–6].
1,1
Definition 4.5. A global solution to the obstacle problem is a positive function w ∈ Cloc
(Rn ) solving
(2.11) in the case A = In e f = 1.
The following result occurs:
Theorem 4.6. Every global solution w is convex. Moreover, if w 6≡ 0 and 2-homogeneous, then one
of the following two cases occurs:
2
(A) w(y) = 12 hy, νi ∨ 0 for some ν ∈ Sn−1 ;
(B) w(y) = hBy, yi with B a symmetric, positive definite matrix satisfying and TrB = 12 .
Having this result at hand, a complete classification of the blow-up limits, for the obstacle problem
(2.7), follows as in the classical context. Focardi, Gelli and Spadaro prove this result in [15, Proposition
4.2 and 4.5] using the quasi-monotonicity formula of Weiss and a Γ-convergence argument:
Proposition 4.7 (Classification of blow-ups). Every blow-up vx0 at a free boundary point x0 ∈ Γu is
of the form vx0 = w(L−1 (x0 )y), with w a non-trivial, 2-homogeneous global solution.
According to Theorem 4.6 we shall call a global solution of type (A) or of type (B).
The above proposition allows us to formulate a simple criterion to distinguish between regular and
singular free boundary points.
Definition 4.8. A point x0 ∈ Γu is a regular free boundary point, and we write x0 ∈ Reg(u) if there
exists a blow-up of u at x0 of type (A). Otherwise, we say that x0 is singular and write x0 ∈ Sing(u).
Remark 4.9. Simple calculations show that Ψw (1) = θ for every global solution of type (A) and
Ψw (1) = 2θ for every global solution of type (B), where Ψw is the energy defined in (3.43) and θ is a
dimensional constant.
16
C 1,γ
Remark 4.10. We observe that for every sequence rj & 0 for which uL(x0 ),rj −−−→ w with w global
solution 2-homogeneous then
lim ΦL(x0 ) (rj ) = Ψw (1).
rj →0
By Weiss’ quasi monotonicity it follows the uniqueness of the limit follows, so ΦL(x0 ) (0) = Ψw (1) for
every w limit of the sequence (uL(x0 ),r )r . It follows that if x0 ∈ Γu is a regular point then ΦL(x0 ) (0) = θ
or, equivalently every blow-up at x0 is of type (A).
4.2
Uniqueness of blow-ups
The last remarks show that the blow-up limits at the free boundary points must be of a unique type:
nevertheless, this does not imply the uniqueness of the limit itself. In this paragraph we prove the
propriety of uniqueness of blow-ups.
In view of Proposition 4.7, if x ∈ Γu the blow-up in x is unique with form
(
2
1
−1
(x)ς(x), yi ∨ 0
x ∈ Reg(u)
2 hL
vx (y) =
−1
−1
hL (x)Bx L (x)y, yi
x ∈ Sing(u).
where ς(x) ∈ Sn−1 is the blow-up direction at x ∈ Reg(u) and Bx is symmetric matrix such that
Tr Bx = 12 .
We start with singular points case. Therefore, by Weiss’ and Monneau’s quasi-monotonicity formulas
it follows that:
Proposition 4.11 ( [15, Proposition 4.11]). For every point x ∈ Sing(u) there exists a unique blow-up
limit vx (y) = w(L−1 (x)y). Moreover, if K ⊂ Sing(u) is a compact subset, then, for every point x ∈ K
uL(x),r − w 1
C (B
1)
≤ σK (r)
∀r ∈ (0, rK ),
(4.4)
for some modulus of continuity σK : R+ → R+ and a radius rK > 0.
Before proceeding with the regular points case, we extend the energy defined in (3.43) for functions
2-homogeneous to each function ξ ∈ W 1,2 (B1 )
ˆ
ˆ
Ψξ (1) =
|∇ξ|2 + 2ξ dx −
ξ 2 dH n−1 .
B1
∂B1
We state a Weiss celebrated epiperimetric inequality [35, Theorem 1] (recently a variational proof for
thin obstacle problem has been given in [16]):
Theorem 4.12 (Weiss’ epiperimetric inequality). There exist δ > 0 and k ∈ (0, 1) such that, for every
ϕ ∈ H 1 (B1 ), 2-homogeneous function, with kϕ − wkH 1 (B1 ) ≤ δ for some global solution w of type (A),
there exists a function ξ ∈ H 1 (B1 ) such that ξ|∂B1 = ϕ|∂B1 , ξ ≥ 0 and
Ψξ (1) − θ ≤ (1 − k) (Ψϕ (1) − θ) ,
(4.5)
where θ = Ψw (1) is the energy of any global solution of type (A).
For convenience of reader we recall the definition (H3’) seen in introduction
(H3)0 Let ω(t) = sup|x−y|≤t |f (x) − f (y)| be the modulus of continuity of f and set a > 2 it holds the
following condition of integrability
ˆ 1
ω(r)
| log r|a dr < ∞.
(4.6)
r
0
In the proof of uniqueness we take advantage of this lemma:
Lemma 4.13. Let u be solution of (2.7) and we assume (H3’) and (1.4). If there exist rays 0 ≤ %0 <
r0 < 1 such that
inf kur |∂B1 − wkH 1 (∂B1 ) ≤ δ
∀ %0 ≤ r ≤ r0 ,
(4.7)
w
where the infimum is taken on all global solutions w of type (A) and δ > 0 is the constant of Theorem
4.12, then for each pair of rays %, t such that %0 ≤ % < t ≤ r0 we have
ˆ
|ut − u% | dH n−1 ≤ C7 ρ(t),
(4.8)
∂B1
with C7 positive constants independent of r and %, while ρ(t) a growing function vanishing in 0.
17
Proof. By the Divergence theorem, (3.18) and (3.33) we can compute the derivative of Φ0 (r) in the
following way:
E 0 (r)
E(r)
H0 (r)
H(r)
−
(n
+
2)
−
2
+ 2(n + 3) n+4
n+3
n+3
rn+2
r
r
r
ˆ
1
8
E(r)
≥ n+2
(hA∇u, ∇ui + 2 f u) dH n−1 − (n + 2) n+3 + n+4 H(r)
r
r
r
∂Br
ˆ
n
4
− n+3
uhAν, ∇ui dH n−1 − C r− Θ(s)
r
∂Br
ˆ
ˆ
1
(n + 2)
2(n − 2)
≥ n+2
(|∇u|2 + 2 u) dH n−1 −
Φ(r) −
u2 dH n−1
r
r
rn+4
∂Br
∂Br
ˆ
n
4
ω(r)
uhν, ∇ui dH n−1 − C r− Θ(s) +
− n+3
r
r
∂Br
ˆ
2
(n + 2)
1
=−
Φ(r) +
hν, ∇ur i − 2ur + |∂τ ur |2 + 2ur − 2n u2r dH n−1
r
r ∂B1
n
ω(r)
,
− C r− Θ(s) +
r
Φ0 (r) =
where we denote by ∂τ ur , the tangential derivative of ur along ∂B1 . Let wr be the 2-homogeneous
extension of ur |∂B1 . We note that if g is a 2-homogeneous function it holds
ˆ
ˆ 1ˆ
ˆ 1
ˆ
g(x) dx =
g(y) dH n−1 (y) dt =
tn+1
g(y) dH n−1 (y)
B1
0
∂Bt
0
∂B1
(4.9)
ˆ
1
g(y) dH n−1 (y).
=
n + 2 ∂B1
Then a simple integration in polar coordinates, thanks to Euler theorem for homogeneous functions
and (4.9) gives
ˆ
ˆ
|∂τ ur |2 + 2ur − 2n u2r dH n−1 =
|∂τ wr |2 + 2wr + 4wr2 − 2(n + 2) wr2 dH n−1
∂B1
∂B1
ˆ
ˆ
2
=
|∇wr | + 2wr − 2(n + 2)
wr2 dH n−1
∂B1
∂B1
ˆ
ˆ
2
n−1
=(n + 2)
(|∇wr | + 2wr ) dH
− 2(n + 2)
wr2 dH n−1 = (n + 2)Ψwr (1).
B1
Therefore, we conclude that
1
(n + 2)
Ψwr (1) − Φ(r) +
Φ (r) ≥
r
r
∂B1
ˆ
0
hν, ∇ur i − 2ur
∂B1
2
dH
n−1
n
ω(r)
− Θ(s)
−C r
+
. (4.10)
r
We can also note that, being wr the 2-homogeneous extension of ur |∂B1 , thanks to (4.9) and (4.7),
there exist a global solution w of type (A) such that
kwr − wkH 1 (B1 ) ≤ √
1
kwr ∂B1 − wkH 1 (∂B1 ) ≤ δ.
n+2
Hence, we can apply the epiperimetric inequality (4.5) to wr and find a function ξ ∈ wr + H01 (B1 )
such that
Ψξ (1) − θ ≤ (1 − k) Ψwr (1) − θ .
(4.11)
Moreover, we can assume without loss of generality (otherwise we substitute ξ with ur ) that Ψξ (1) ≤
Ψur (1). Then, by the minimality of ur in E with respect to its boundary conditions (1.4), (H1)-(H3)
and lemma 3.1 we have that
ˆ
ˆ
Ψξ (1) =
|∇ξ|2 + 2ξ dx −
ξ 2 dH n−1
∂B1
ˆB1
ˆ
≥
hA(rx)∇ξ, ∇ξi + 2 f (rx)ξ dx −
µ(rx) ξ 2 dH n−1
B1
∂B1
ˆ
ˆ
1− pn∗
2
(4.12)
−C r
+ ω(r)
|∇ξ| + 2ξ dx − C rγ
ξ 2 dH n−1
B1
∂B1
ˆ
ˆ
n
|∇ξ|2 + 2ξ dx − C rγ
ξ 2 dH n−1
≥Φ(r) − C r1− p∗ + ω(r)
∂B1
B1
n
≥Φ(r) − C r1− p∗ + ω(r) .
18
By (4.11) and (4.12) we get
Ψwr (1) − Φ(r) ≥
n
1
k
(Φ(r) − θ − C rβ ) + θ − Φ(r) =
(Φ(r) − θ) − C r1− p∗ + ω(r) . (4.13)
1−k
1−k
Then by (4.10) and (4.12)
Φ0 (r) ≥
e6 ∈ (0, (1 −
Let now C
n
Θ(s) )
n+2 k
(Φ(r) − θ) − C
r 1−k
n
r− Θ(s) +
ω(r)
r
.
(4.14)
k
∧ (n + 2) 1−k
), then
(Φ(r) − θ) r
e6
−C
0
≥ −C
r
n
e6
− Θ(s)
−C
+
ω(r)
r1+Ce6
.
(4.15)
Indeed, by taking into account (4.14)
e
e 0
e6 (Φ(r) − θ) r−Ce6 −1
(Φ(r) − θ) r−C6 = Φ0 (r)r−C6 − C
n
n+2 k
ω(r)
e
− Θ(s)
e6 (Φ(r) − θ) r−Ce6 −1
≥
+
(Φ(r) − θ) − C r
r−C6 − C
r 1−k
r
n
k
ω(r)
e6 −1
e
− Θ(s)
−C
e
≥ (Φ(r) − θ)r
(n + 2)
− C6 − C r
+
r−C6
1−k
r
n
ω(r)
e6
− Θ(s)
−C
≥ −C r
.
+
r1+Ce6
By integrating (4.15) in (t, r0 ) with t ∈ (s0 , r0 ) and multiplying by tC6 we finally get
r0
ˆ r0 n
ω(r)
e6
e6
e6
e6
−C
− Θ(s)
C
−C
C
+
t
(Φ(r) − θ) r
≥ −C t
r
dr
r1+Ce6
t
t
e
whence
ˆ
r0
n
ω(r)
e6
e
− Θ(s)
−C
Φ(t) − θ ≤ C
r
+
dr + 1 tC6
e6
1+
C
r
t
ˆ r0
ˆ r0
n
ω(r)
ω(r)
e
e
e6
1− Θ(s)
C6
C6
C
≤C r
+t +t
dr ≤ C t
dr + 1 .
r1+Ce6
r1+Ce6
t
t
(4.16)
Consider now %0 < % < r0 and estimate as follows
ˆ t
ˆ
d u(rx)
n−1
dr dH n−1
|ut −u% | dH
=
2
r
∂B1
∂B1
% dr
ˆ t
ˆ ˆ t
ˆ
h∇u(rx), xi − 2 u(rx) dH n−1 dr = r−1
≤ r−2
|h∇ur (x), xi − 2 ur (x)| dH n−1 dr
r
%
∂B1
∂B1
%
12
ˆ t
ˆ
√
1
2
r− 2 r−1
|h∇ur (x), xi − 2 ur (x)| dH n−1
≤ nωn
dr.
ˆ
%
∂B1
Combining (3.29), (4.10), (4.13), (4.16) and Hölder inequality we have
ˆ
ˆ
21
n
1
ω(r)
r− 2 Φ0 (r) + C r− Θ(s) +
dr
r
%
1 12
ˆ t
n
n
t 2
ω(r)
1− Θ(s)
1− Θ(s)
≤ C log
Φ(t) − Φ(%) + C t
−%
+
dr
%
r
%
1 12
ˆ t
n
t 2
ω(r)
1− Θ(s)
(Φ(t) − θ) + (θ − Φ(%)) + C t
≤ C log
+
dr
%
r
%
1 12
ˆ r0
ˆ t
t 2
ω(r)
ω(r)
e6
e6
C
C
≤ C log
t +t
dr + ω(t) +
dr
.
%
r
r1+Ce6
t
%
t
|ut − u% | dH n−1 ≤ C
∂B1
19
(4.17)
Now thanks to the hypothesis (H3)0 , if r0 << 1 for every 0 ≤ t ≤ r0 we can apply the inequality
e
ω(t) ≤ | log t|−a and the infinitesimal function tC6 | log t|a is growing, taking (4.17) into account and
decreasing of | log t|a we have
1
12
1
ˆ r0
a
a
ω(r) | log r|a
t 2
t 2
| log t|− 2 1 +
dr
≤ C log
| log t|− 2 .
|ut − u% | dH n−1 ≤ C log
%
r
%
t
∂B1
(4.18)
ˆ
A simple dyadic decomposition argument then leads to the conclusion. If % ∈ [2−k , 2−k+1 ) and
t ∈ [2−h , 2−h+1 ) with h < k, applying (4.18)
ˆ
|ut − u% | dH n−1 ≤ C
∂B1
k
X
a
log(2j )− 2 ≤ C7
j=h
with
ρ(t) :=
∞
X
1
a
2
j
j=h
∞
X
1
a =: C7 ρ(t),
j2
j=h
if t ∈ [2−h , 2−h+1 ).
(4.19)
By taking (1.5) into account we have a > 2, therefore, the function ρ(t) is growing and infinitesimal in
0, from which the conclusion of the lemma follows.
Checking the hypothesis of Lemma 4.13 it’s possible to prove the uniqueness of the blow-ups at
regular points of the free boundary:
Proposition 4.14 ( [15, Proposition 4.10]). Let u be a solution to the obstacle problem (2.11) with f
that satisfies (H3)0 and x0 ∈ Reg(u). Then, there exist constants r0 = r0 (x0 ), η0 = η0 (x0 ) such that
every x ∈ Γu ∩ Bη0 (x0 ) is a regular point and, denoting by vx = w(L−1 (x)y) any blow-up of u in x we
have
ˆ
|uL(x),r − w| dH n−1 (y) ≤ C7 ρ(r)
∀ r ∈ (0, r0 ),
(4.20)
∂B1
where C7 is an independent constant from r and ρ(r) a growing, infinitesimal function in 0. In
particular, the blow-up limit vx is unique.
Remark 4.15. If f is α-Hölder we can prove Lemma 4.13 and Proposition 4.14 with ρ(t) = tC6 where
C6 := C̄62∧α .
4.3
Regularity
In this last section we state some regularity results of free boundary for u solution of (2.7). If the
matrix A satisfies the hypotheses (H1)-(H2) and the linear term f satisfies the hypothesis (H3)0 we
obtain differentiability of the free boundary in a neighborhood of any point x ∈ Reg(u). In particular
if f is Hölder we establish the C 1,β regularity as in [15] where A is Lipschitz continuous.
Since the arguments involved are those used in [15] togheter with the preliminaries developed in
the previous sections we do not provide any proof.
Theorem 4.16 ( [15, Theorem 4.12]). Assume hypotheses (H1), (H2) and (H3)0 hold. Let x ∈ Reg(u).
Then, there exists r > 0 such that Γu ∩ Br (x) is hypersurface C 1 and n its normal versor is absolutely
continuous with modulus of continuity depending on ρ defined in (4.19).
In particular if f is Hölder continuous there exists r > 0 such that Γu ∩ Br (x) is hypersurface C 1,β
for some universal exponent β ∈ (0, 1).
We are able to say less on the set of singular points we are able to say less. We know that below
the hypotheses (H1)-(H3), the set Sing(u) is contained in union of C 1 submanifold.
Definition 4.17. The singular stratum Sk of dimension k for k = 0, 1, . . . , n − 1 is the subset of points
x ∈ Sing(u) for which Ker(Bx ) = k.
In the following theorem we show that Sk is H k rectifiable, and moreover that ∪n−1
k=l Sk is a closed
set for every l = 0, 1, . . . , n − 1.
Theorem 4.18 ( [15, Theorem 4.14]). Assume hypotheses (H1), (H2) and (H3). Let x ∈ Sk . Then
there exists r such that Sk ∩ Br (x) is contained in regular k-dimensional submanifold of Rn .
20
Acknowledgments
The author is warmly grateful to Professor Matteo Focardi for the suggestion of the problem and for
his constant support and encouragement. The author is partially supported by project GNAMPA 2015
“Regolarità per problemi di analisi geometrica e del calcolo delle variazioni”. The author is member of
the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of
the Istituto Nazionale di Alta Matematica (INdAM).
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