OPTIMAL THREE CYLINDER INEQUALITIES FOR SOLUTIONS
TO PARABOLIC EQUATIONS WITH LIPSCHITZ LEADING
COEFFICIENTS.
LUIS ESCAURIAZA AND SERGIO VESSELLA
Abstract. We prove a Carleman estimate with a singular weight for parabolic
operators P with time-dependent and Lipschitz second order coefficients. As
a consequence we obtain a three-cylinder inequality with optimal exponent for
solutions to the parabolic inequality |P u| ≤ C (|∇u| + |u|) .
2000 Mathematical Subject Classification. 35B60, 35R25, 35K10
1. Introduction
The Carleman inequality
(1.1)
−α −α+2
∆u
|x| u 2 n ≤ C |x|
L (R )
L2 (Rn )
, α∈
/
n
n
+ N , u ∈ C∞
0 (R \ {0}) ,
2
can be used to show that the so called ”three-sphere inequality” holds for harmonic
functions [7, Theorem 2]:
There is a constant C = C(n) such that, the following inequality holds when
1
ρ < r ≤ and u is a harmonic function in B1
2
C
1
log 2r
C
θ
1−θ
kukL2 (Bρ ) kukL2 (B1 ) , where θ =
(1.2)
kukL2 (Br ) ≤
r
log ρ1
and Br = {x ∈ Rn : |x| < r}.
A similar statement can be proved for solutions to elliptic equations with Lipschitz second order coefficients, replacing the Carleman inequality (1.1) by a suitable
analog. Moreover, the three-sphere inequality is well known to imply that nonconstant solutions to elliptic equations cannot have a zero of infinite order at interior
points and have applications to obtain stability estimates for elliptic inverse problems with unknown boundary [1].
In the context of parabolic operators
(1.3)
P u = ∂i g ij (x, t) ∂j u − ∂t u ,
the proper analog of (1.2) would be the following ”two-sphere one-cylinder inequality”:
There is a constant C > 1 such that, the following inequality holds when ρ <
r ≤ 1/C and u verifies P u = 0 in B1 × (−1, 0]
(1.4)
C
1
log Cr
C
θ
1−θ
ku (., 0)kL2 (Br ) ≤
ku (., 0)kL2 (Bρ ) kukL2 (B1 ×(−1,0]) , where θ =
.
r
C log ρ1
1
2
LUIS ESCAURIAZA AND SERGIO VESSELLA
This inequality has been established for parabolic operators P with time independent coefficients (i.e. g ij (x, t) = g ij (x)), under the condition that the matrix of
n
coefficients g ij (x) i,j=1 is Lipschitz continuous [3]. The arguments used by these
authors, reduce (1.4) to the previously established elliptic counterparts, adapting to
this case the techniques used in [6] and [7] to prove that if u is a solution to P u = 0
in B1 × (−1, 0] and u (., 0) is nonconstant, then u (., 0) cannot have a zero of infinite
order at x = 0. Those arguments rely on a representation formula for solutions to
parabolic equations in terms of eigenfunctions of the corresponding elliptic operator and therefore, cannot be applied to parabolic equations with time-dependent
coefficients.
Though the previous unique continuation property has been established in [2]
and [5] for time-dependent parabolic operators when the matrix of coefficients lies
1
in C 1, 2 (Rn+1 ), it is not clear that the arguments used in the latter works imply
(1.4) and it remains unknown.
In this note we prove an optimal ”three-cylinder inequality” for solutions to
parabolic inequalities with time dependent coefficients which can be applied to obtain sharp stability estimates for inverse parabolic problems with unknown timeindependent boundary. In particular, in Theorem 1 below we set Qtr = Br × (−t, t)
and assume that the symmetric matrix of coefficients of P verifies, for some Λ ≥ 1,
the following conditions when ξ ∈ Rn and (x, t), (y, s) ∈ Rn+1
2
Λ−1 |ξ| ≤
(1.5)
n
X
2
g ij (x, t) ξi ξj ≤ Λ |ξ|
,
i,j=1
(1.6)
n
X
ij
g (x, t) − g ij (y, s) ≤ Λ (|x − y| + |t − s|) .
i,j=1
Theorem 1. Assume that u ∈ H 2,1 Q11 satisfies
∂i g ij (x, t) ∂j u − ∂t u ≤ Λ (|∇u| + |u|)
in Q11 .
Then, there exists a constant C = C(n, Λ) > 1 such that, the following inequality
holds when R, T ∈ (0, 1], ρ < r < R/C and τ ∈ (0, T ),
β
CT R
1−θ
θ
kukL2 (QT ) kukL2 (QT ) ,
(1.7)
kukL2 (QrT −τ ) ≤
ρ
R
τr
where
θ=
R
log Cr
C log R
ρ
and
β=C
T
R2
+
T
τ
C
.
The second author has previously proved this three-cylinder inequality
[8] under
the hypotheses that the matrix of coefficients of P lies in C 2,1 Rn+1 . In order
to obtain Theorem 1, the main effort in [8] and in this work is to find a suitable
generalization for parabolic operators with variable coefficients of the following
Carleman estimate for the heat operator, Assume that u ∈ C0∞ Rn+1 \ {0} × R . Then, the following inequalities hold
when α ≥ 1
√
(1.8)
αk|x|−α−1 (x · ∇u + 12 (n − 2 − 2α)u)kL2 (Rn+1 ) + √12α k|x|1−α ∂t ukL2 (Rn+1 )
≤ k|x|1−α (∆u − ∂t u) kL2 (Rn+1 ) .
OPTIMAL THREE CYLINDER INEQUALITIES
3
In section 2 we prove the generalization of this Carleman inequality for parabolic
operators with variable coefficients. The Carleman inequality (1.8) follows setting
w = |x| and g ij (x, t) = δij in the Lemmas 1 and 2 below. We omit the proof of the
three-cylinder inequality (1.7) because it is the same as the one given in [8] with
the obvious changes.
2. Main Results
In what follows dX = dxdt and I is the identity matrix.
Theorem 2. Assume that the parabolic operator P satisfies the conditions (1.5)
and (1.6). Then, there are constants 0 < δ < 1 and N > 1 depending on n and Λ,
and a function w = w(x, t) satisfying |x|/N ≤ w(x, t) ≤ N |x| on Q11 such that, the
following inequalities hold for all α ≥ N and u ∈ C0∞ Q1δ \ {0} × R
Z
Z
2
2
(2.1)
αw1−2α |∇u| + α3 w−1−2α u2 dX ≤ N w2−2α (P u) dX .
To simplify the writing and calculations in the proof of Theorem 2 we shall use
some of the standard notations in Riemannian geometry, but always dropping the
corresponding volume element in the definition of the Laplace-Beltrami operator
associated to a Riemannian metric. We do this, because it simplifies the formulae
appearing in the proofs of the following lemmas, and especially when the metric is
allowed to depend on the time variable and we make use of partial integration with
respect to this variable.
n
In particular, letting g (x, t) = {gij (x, t)}i,j=1 to denote the inverse matrix of
n
the matrix of coefficients of P , we have g −1 (x, t) = g ij (x, t) i,j=1 , and we use
the following notations when considering either a function f or two variable vector
fields ξ and η:
Pn
2
1. ξ · η = i,j=1 gij (x, t) ξi ηj , |ξ| = ξ · ξ.
−1
2. ∂ f = ∂f , ∂t f = ∂f
∇x f , div (ξ) =
∂t , ∇x f = (∂1 f, . . . , ∂n f ) , ∇f = g
Pn i i ∂xi
∂
ξ
and
∆f
=
div
(∇f
).
i=1 i
With this notation the following formulae hold when u, f and h are smooth
functions
2
(2.2)
P u = ∆u + ∂t u , ∆ f 2 = 2f ∆f + 2 |∇f |
and
Z
(2.3)
Z
f ∆hdx =
Z
h∆f dx = −
∇f · ∇hdx .
The arguments we use to prove Theorem 2 follow the standard methods. First,
we set f = w−α u and compute w−α P u in terms of f ,
2
2
|∇w|
|∇w|
f − α (∂t logw) f + 2α
A (f ) − ∂t f = Lα f .
2
w
w2
Then, we split (w/|∇w|)Lα into its symmetric and antisymmetric parts, and choose
the function w in such a way that the commutator of the latter operators is a positive
operator. In order to calculate the quadratic form associated to the commutator we
use partial integration and the Rellich-Nečas identity with vector field w∇w/|∇w|2 .
This calculation is done in Lemma 1. The Lemma 2 is used to control from below
(2.4) w−α P u = ∆f + α2
4
LUIS ESCAURIAZA AND SERGIO VESSELLA
the quadratic form associated to the commutator ant to justify our choice of the
function w in (2.16) .
In Lemma 1 below, the reader should have in mind that w denotes a function in
C 2,1 (Q11 ) verifying |x|/C ≤ w ≤ C|x| and |∇w| > 0 in Q11 , f = w−α u,
(2.5)
A (f ) = w
2
∇w · ∇f
1
w∆w − |∇w|
+ Fwg f , Fwg =
2
2
|∇w|
2
|∇w|
,
and Mgw denotes the n × n symmetric matrix
1
n
{Mij + Mji }i,j=1 ,
2
where using the summation notation of repeated indices
!
!
1
1
1
wg ik ∂k w
wg kl ∂l w
w∇w
hi
δij − ∂j
+ gjh
− Fwg δij .
Mij = div
2
2
2 ∂k g
2
2
2
|∇w|
|∇w|
|∇w|
Mgw =
(2.6)
In the sequel all the integrals are done over Q11 and with respect to Lebesque
measure dX.
Lemma 1. Assume that w ∈ C 2,1 (Q11 ) satisfies the above properties. Then, the
following inequality holds for all α ≥ 1 and u ∈ C0∞ (Q11 \ {0} × R)
Z
Z
Z
Z
2
w2
−α
g
g
2
(2.7)
P u ≥ 4α Mw ∇f · ∇f + α Fw ∆ f − 2 Fwg f ∂t f
2 w
|∇w|
Z
Z
Z
2
1
w2
w2
|∇w|
2
2
2
A
(f
)
+
(∂
f
)
−
|∇f | ∂t
+2α
t
2
2
2
w
α
|∇w|
|∇w|
Z 2
Z
2
w ∇ |∇w| · ∇f
w2
ij
∂
g
∂
f
∂
f
−
2
∂t f
−
t
i
j
2
4
|∇w|
|∇w|
Z
Z
w2
2
−4α
(∂t log w) A (f ) f + 2α (∂t log w)
2 f ∂t f .
|∇w|
Proof. Defining
Z J1 =
2α
|∇w|
w
A (f ) −
∂t f
w
|∇w|
2
and
Z
J2 = 2
2αA (f ) −
w2
!
2 ∂t f
|∇w|
∆f +
2
|∇w|
α2
f
w2
!
− α (∂t log w) f
we get from (2.4)
Z
(2.8)
w2
2
|∇w|
w−α P u
2
≥ J1 + J2 .
We begin studying the integral J2 . To this end, the observation
Z
2
|∇w|
(2.9)
A (f ) f = 0 ,
w2
the identity
2
2∇f · ∇∂t f = ∂t |∇f | − ∂t g ij ∂i f ∂j f
,
OPTIMAL THREE CYLINDER INEQUALITIES
5
and the divergence theorem give the formula
Z
Z
Z
w2
w2
w2
(2.10)
−2 ∂t f ∆f
∇f · ∇∂t f
∂t f ∇f · ∇
2 =2
2 +2
2
|∇w|
|∇w|
|∇w|
Z
Z
w2
w2
2
ij
−
= − |∇f | ∂t
2
2 ∂t g ∂i f ∂j f
|∇w|
|∇w|
Z
Z 2
2
w∇w · ∇f
w ∇ |∇w| · ∇f
+4
∂
f
−
2
∂t f .
t
2
4
|∇w|
|∇w|
From (2.5) we have
Z
Z
Z
w∇w · ∇f
(2.11)
4
∂
f
=
4
A
(f
)
∂
f
−
2
Fwg f ∂t f .
t
t
2
|∇w|
Then, (2.9), (2.10) and (2.11) and the identity
Z
f ∂t f = 0
imply that
Z
(2.12)
J2 = 4α
Z
+4
Z
A (f ) ∂t f − 2
Z
A (f ) ∆f −
Fwg f ∂t f − 2
Z
2
|∇f | ∂t
w2
Z
2
|∇w|
−
w2
2
w2 ∇ |∇w| · ∇f
2 ∂t g
|∇w|
∂t f − 4α2
ij
∂i f ∂j f
Z
(∂t log w) A (f ) f
4
|∇w|
Z
w2
+2α (∂t log w)
2 f ∂t f .
|∇w|
2
Now, from (2.5) and the identity ∆ f 2 = 2f ∆f + 2 |∇f | we get
Z
Z
Z
w∇w · ∇f
2
g
(2.13)
4 A (f ) ∆f = 2 2
∆f − Fw |∇f | + Fwg ∆ f 2 .
2
|∇w|
Then, the following Rellich-Nečas (or Pokhozaev) identity
2
2
2 (β · ∇f ) ∆f = 2div ((β · ∇f ) ∇f ) − div β |∇f | + div (β) |∇f |
−2∂i β k g ij ∂j f ∂k f + β k ∂k g ij ∂i f ∂j f
w∇w
with vector field β =
2 , (2.13) and the divergence theorem give the identity
|∇w|
Z
Z
Z
(2.14)
4 A (f ) ∆f = 4 Mgw ∇f · ∇f + Fwg ∆ f 2 ,
where Mgw is defined by (2.6). Now, plugging (2.14) into (2.12) we obtain from
(2.17) the following inequality
Z
Z
Z
2
w2
−α
g
P u ≥ 4α Mw ∇f · ∇f + α Fwg ∆ f 2
(2.15)
2 w
|∇w|
2
Z |∇w|
w
+
2α
A (f ) −
∂t f
+ 4A (f ) ∂t f
w
|∇w|
Z
Z
Z
w2
w2
2
ij
− |∇f | ∂t
−
∂
g
∂
f
∂
f
−
2
Fwg f ∂t f
i
j
2
2 t
|∇w|
|∇w|
6
LUIS ESCAURIAZA AND SERGIO VESSELLA
Z
−2
2
w2 ∇ |∇w| · ∇f
∂t f − 4α
2
Z
Z
(∂t log w) A (f ) f + 2α
(∂t log w)
w2
2 f ∂t f .
|∇w|
|∇w|
On the other hand, using elementary algebra it is simple to show that when α ≥ 1
2
2
1 w2
|∇w|
w
|∇w|
2
2
A
(f
)
+
2α
A (f ) −
∂t f
+ 4A (f ) ∂t f ≥ 2α
(∂t f ) .
w
|∇w|
w2
α |∇w|2
4
This inequality bounds from below the third integral on the right hand side of (2.15)
by
Z
Z
2
1
w2
|∇w|
2
2
A
(f
)
+
,
2α
2 (∂t f )
w2
α
|∇w|
and Lemma 1 follows from this fact and (2.15).
In the following lemma we state the main properties of the function Fwg and
matrix Mgw which we use in the proof of Theorem 4. We will omit the proof
of the lemma because it can be achieved with the chain rule and straightforward
calculations.
Lemma 2. Let ϕ be a positive non-decreasing function on (0, +∞) and define

1/2
n
X
φ (s) = ϕ (s) / (sϕ0 (s)) and σ (x, t) = 
gij (0, t) xi xj 
.
i,j=1
Then, the symmetric matrix
Mgw
= 0, and the following facts hold
!
∇w
⊗
∇w
= φ (w) Mgw + wφ0 (w) I −
g
2
|∇w|
verifies
g
Fϕ(w)
= φ (w) Fwg − wφ0 (w) , Mgϕ(w)
Mgw ∇w
Fσg(0,t) = n − 2 , Mg(0,t)
=0 .
σ
For a fixed number µ ≥ 1 to be chosen later, we define
Z
(2.16)
w (x, t) = ϕ (σ (x, t)) , where ϕ(s) = s exp
0
s
e−µτ − 1
dτ
τ
and σ is the function defined in Lemma 2. Here the reader
Pn should observe that σ is
the distance function to x = 0 associated to the metric i,j=1 gij (0, t)dxi dxj . With
these choices, φ(s) = eµs , and the reader can verify that the following properties
hold on Q11 for some constant C depending on n, Λ and µ
(2.17) σC ≤ w ≤ Cσ , 1/C ≤ |∇w| ≤ C , |∂t (w2 /|∇w|2 )| ≤ Cw2 , |∂t φ| ≤ Cσ
2
|∇|∇w| | ≤ C , |∆φ| ≤ Cw−1 , |Fwg | ≤ C and |∂t log w| ≤ C .
Now we estimate from below the sum of the three first integrals on the right-hand
side of (2.7). From Lemma 1
!
2
(∇σ · ∇f )
2
g
0
Mw ∇f · ∇f = σφ |∇f | −
+ φMgσ ∇f · ∇f ,
2
|∇σ|
e the tangential components of the gradient of f along the level
and denoting as ∇f
Pn
sets of σ(x, t) with respect to the metric i,j=1 gij (x, t)dxi dxj , we have
e = ∇f − (∇σ · ∇f ) ∇σ = ∇f − (∇w · ∇f ) ∇w .
∇f
2
2
|∇σ|
|∇w|
OPTIMAL THREE CYLINDER INEQUALITIES
7
g(0,t)
Now, from Lemma 2 Mgσ ∇σ = 0 and Mσ
= 0, and these two facts imply that
e · ∇f
e .
Mgσ ∇f · ∇f = Mgσ − Mg(0,t)
∇f
σ
On the other hand, a calculation and the Lipschitz condition (1.6) give that there
is a constant N depending on n and Λ such that
g
≤ Nσ ,
Mσ − Mg(0,t)
σ
Z
(2.18)
Mgw ∇f · ∇f ≥
Z
e |2 .
σ (φ0 − N φ) |∇f
In order to estimate from below the second and third integrals on the right-hand
side of (2.7) we observe that from Lemma 1
Fwg = (n − 2) φ + (Bφ − σφ0 ) ,
(2.19)
g(0,t)
where B = Fσg − Fσ
satisfies with the same constant N as above
|B (x, t)| ≤ N σ .
(2.20)
From the identities
2
2
2
e |2 + (∇w · ∇f ) , f ∂t f = 1 ∂t f 2 ,
∆ f 2 = 2f ∆f + 2 |∇f | , |∇f | = |∇f
2
2
|∇w|
(2.19) and the divergence theorem we have
Z
Z
Z
(2.21)
Fwg ∆ f 2 = (n − 2) f 2 ∆ (φ) + 2 (Bφ − σφ0 ) f ∆f
Z
2
2
e | +2
(Bφ − σφ0 ) |∇f
Z
2
(Bφ − σφ0 )
(∇w · ∇f )
,
2
|∇w|
and
Z
(2.22)
Fwg f ∂t f = −
(n − 2)
2
Z
f 2 ∂t φ +
Z
(Bφ − σφ0 ) f ∂t f .
Concerning the second integral on the right-hand side of (2.21), the identity (2.4)
gives the formula
Z
(2.23)
2 (Bφ − σφ0 ) f ∆f =
Z
=2
(Bφ − σφ0 ) w−α f P u + 2α2
Z
2
(σφ0 − Bφ)
2
Z
(σφ0 − Bφ) f
+2
2α
|∇w| 2
f
w2
!
|∇w|
A (f ) − ∂t f − α (∂t log w) f
w2
.
Then, from (2.18), (2.20), (2.21), (2.22) and (2.23) we obtain that the following
inequality holds when α ≥ 1
Z
Z
Z
g
g
2
(2.24)
4α Mw ∇f · ∇f + α Fw ∆ f − 2 Fwg f ∂t f ≥
Z
2α
e |2 + 2α3
(σφ − 2N σφ + Bφ) |∇f
0
Z
2
(σφ0 − Bφ)
|∇w| 2
f − R1 ,
w2
8
LUIS ESCAURIAZA AND SERGIO VESSELLA
where from (2.17) and (2.20), there is a constant C depending on n, Λ and µ such
that
Z
Z
Z
2
(∇w · ∇f )
1−α
2
(2.25)
R1 /C ≤ α w
|f P u| + α w
+α
w−1 |f A (f ) |
|∇w|2
Z
Z
+α w |f ∂t f | + α2 f 2 .
Here we choose µ = 4N , and from (2.7), (2.20), (2.24), (2.25) and (2.17) we find
that for some new constant N depending on n and Λ
Z
Z
Z
Z
e |2 + α3 w−1 f 2 + α w−2 A (f )2 + α−1 w2 (∂t f )2
(2.26)
α w|∇f
Z
≤N
2
w2−2α (P u) + R2 ,
where from (2.7), (2.17) and (2.25)
Z
Z
Z
2
(∇w · ∇f )
2
(2.27)
R2 /N ≤ α w1−α |f P u| + α w
+
α
w−1 |f A (f ) |
2
|∇w|
Z
Z
Z
Z
2
2
2
2
+α w |f ∂t f | + α
f + w |∇f ∂t f | + w2 |∇f | .
Recalling the definition (2.5) of A(f ) and that we claimed that |Fwg | ≤ N in
(2.17), it follows that
2
Z
(2.28)
w
(∇w · ∇f )
2
|∇w|
Z
≤N
2
w−1 A (f ) + N
Z
w−1 f 2 ,
and from (2.26), (2.27) and (2.28)
Z
Z
Z
Z
2
2
2
(2.29)
α w|∇f | + α3 w−1 f 2 + α w−2 A (f ) + α−1 w2 (∂t f )
Z
≤N
2
w2−2α (P u) + R3 ,
where
Z
R3 /N ≤ α
(2.30)
+α2
Z
w−1 |f A (f ) | + α
w1−α |f P u| + α
Z
w |f ∂t f | + α2
Z
Z
w−1 f 2 + α
f2 +
Z
Z
2
w−1 A (f )
w2 |∇f ∂t f | +
Z
2
w2 |∇f |
.
Finally, recalling that f = w−α u, it is simple to show with the Cauchy-Schwarz’s
inequality that all the integrals on the right hand side of (2.30) can be hidden on
the left hand side of (2.29), when u ∈ C0∞ (Q1δ \ {0} × R) and δ ∈ (0, 1), α ≥ 1 are
respectively sufficiently small and large depending on n and Λ. These prove the
Carleman inequality in Theorem 2.
Acknwoledgements. The research of L. Escauriaza was supported by Spanish
Government grant BFM 2001-0458 and by the European Commission via the network Harmonic Analysis and Related Problems, project number RTN2-2001-00315..
The research of S. Vessella was supported by MIUR grant number MM01111258.
OPTIMAL THREE CYLINDER INEQUALITIES
9
References
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Luis Escauriaza, Universidad del Paı́s Vasco-Euskal Herriko Unibertsitatea, Bilbao, Depto. Matemáticas, 48080 Bilbao, Spain. E-mail: [email protected]
Sergio Vessella, DiMaD, Universita’ di Firenze, via Lombroso 6/17, 50134 Firenze,
Italy E-mail: [email protected]