A stability result in the localization of cavities in a thermic conducting medium

URL:http://www.emath.fr/cocv/ DOI: 10.1051/cocv:2002066

A STABILITY RESULT IN THE LOCALIZATION OF CAVITIES IN A THERMIC CONDUCTING MEDIUM

B. Canuto

, Edi Rosset

and S. Vessella

Abstract. We prove a logarithmic stability estimate for a parabolic inverse problem concerning the localization of unknown cavities in a thermic conducting medium Ω in^Rn,n≥2, from a single pair of boundary measurements of temperature and thermal flux.

Mathematics Subject Classification. 35R30, 35R25, 35R35.

Received September 27, 2001. Revised May 28, 2002.

1. Introduction and the main result

In the present paper we are concerned with the study of a problem in thermal imaging. This is a technique used to determine some physical and geometrical proprieties of a thermic conducting medium via boundary measurements of temperature and thermal flux. More precisely we denote by Ω a thermic conducting medium, i.e. a sufficiently smooth, bounded domain inRⁿ,n≥2, and byDacavityin Ω (i.e. Dis a domain compactly contained in Ω), of which neither the form nor the position is known. On the other hand we can measure thetemperature f and the thermal flux g on the boundary of the medium ∂Ω. The goal is then to identify the cavityD via the boundary data f, g. This problem can occur in nondestructive tests of materials, for example in detecting the corrosion parts of an aircraft which are inaccessible to direct inspections (see Bryan and Caudill [5–7], and their references).

We denote byu(t, x) the temperature at the time tand at the pointx∈Ω\D,u0 the initial temperature in Ω\D,f the temperature on (0, T)×∂Ω, andk(x) the anisotropic thermal diffusion coefficient, that is kis an n×nsymmetric matrix-valued function in Ω satisfying the following conditions:

(i) there exists a constantλ≥1, such that for all x∈Ω, and for allξ∈Rⁿ,

λ⁻¹|ξ|²≤k(x)ξ·ξ≤λ|ξ|² (ellipticity), (1.1)

Keywords and phrases:Parabolic equations, strong unique continuation, stability, inverse problems.

1Laboratoire de Math´ematiques Appliqu´ees, UMR 7641 du CNRS, Universit´e de Versailles, 45 avenue des ´Etats-Unis, 78035 Versailles Cedex, France; e-mail:[email protected]

2Dipartimento di Scienze Matematiche, Universit`a degli Studi di Trieste, Via Valerio 12/1, 34100 Trieste, Italy;

e-mail:[email protected]

3DIMAD, Universit`a degli Studi di Firenze, Via C. Lombroso 6/17, 50134 Firenze, Italy; e-mail:[email protected] c EDP Sciences, SMAI 2002

(ii) there exists a constant Λ≥0, such that for allx,y∈Ω,

|k(x)−k(y)| ≤Λ|x−y| R0

(Lipschitz continuity), (1.2)

where R0 is a positive constant related to the size of Ω (see Th. 1.1 and Sect. 2 below for a precise definition).

For Ω, D, k, u0, f assigned, suppose that u solves the following parabolic problem, which we call the direct problem:















ut−div(k(x)∇u) = 0 in (0, T)×Ω\D, u(0) =u0 in Ω\D,

u= 0 on (0, T)×∂D, u(t, σ) =f(t, σ) on (0, T)×∂Ω.

(1.3)

It is well-known that, under reasonable assumptions on the data, problem (1.3) has a unique solution, and that the thermal flux

k(σ)∇u(t, σ)·n(σ)

is well-defined for (t, σ)∈(0, T)×∂Ω. (Here and in the sequeln(σ) denotes the exterior unit normal atσ∈∂Ω.) In the present paper we are interested in the following two problems:

(a) uniquenessresult: for anyu0,f assigned in (1.3), does the thermal fluxk∇u·n_|(0,T)×Γ on (0, T)×Γ of the corresponding solutionudetermineuniquelythe domainD in Ω?

(b) stability result: for any u0 and f assigned in (1.3), does D depend continuously on the thermal flux k∇u·n_|(0,T)×Γ?

Here and in the sequel Γ denotes a relatively open piece of∂Ω.

We begin by observing that, following a counterexample of Bryan and Caudill [6], uniqueness result (a) can fail without additional hypotheses on the datau0, f. In fact letD1, D2 be the following two rectangles in R²: D1 := (0, π)×(0,2π), D2 := (0, π)×(0, π), and let Ω be a bounded domain in R² containing D1. For u(t, x1, x2) := e^−2tsinx1sinx2, let us define the functions u1, u2 as follows: u1 := u_{|(0,T)×Ω\D1}, u2 :=

u_{|(0,T)×Ω\D2}. It is clear thatu1,u2are solutions of (1.3), respectively whenD:=Di,i= 1,2,k(x) :=I2(I2is the 2×2 identity matrix). Moreover _∂n^∂ u1= _∂n^∂ u2on (0, T)×∂Ω. So in this case uniqueness fails.

On the other hand if we assume that in (1.3) the initial temperatureu0isconstant(but a priori unknown), then it is not difficult to prove uniqueness result (a) for any datum f ∈ C¹([0, T], H¹²(∂Ω)), f 6≡0. In fact suppose that there exist two domainsD1, D2 (here and in the sequel Ω\Di is supposed connected) and two constantsc1,c2such that the corresponding solutionsui∈C((0, T), H¹(Ω\Di))∩C¹([0, T], L²(Ω\Di)) of (1.3), whenD:=Di, and the initial temperatureui0≡ci, have the same thermal flux on (0, T)×Γ, that is

k∇u1(t)·n_|Γ=k∇u2(t)·n_|Γ inH⁻¹²(Γ) for allt∈(0, T).

We denote byGthe connected component of Ω\(D1∪D2) such that∂Ω⊂∂G. Let us define u:=u1−u2 in (0, T)×G.

Thenusolves









ut−div(k(x)∇u) = 0 in (0, T)×G, u= 0 on (0, T)×∂Ω, k∇u·n= 0 on (0, T)×Γ.

By the unique continuation principle (see Lin [15]) it follows thatu≡0 in [0, T)×G, that is

u1=u2 in [0, T)×G. (1.4)

This in particular implies thatc1=c2. Next let us denote by vi :=uit in [0, T)×Ω\Di.

Let assume, for instance, thatD2\D16=∅. We have that (Ω\D1)\G6=∅, andv1 solves















v1t−div(k(x)∇v1) = 0 in (0, T)×Ω\D1, v1(0) = 0 in Ω\D1,

v1 = 0 on (0, T)×∂D1, v1 =fton (0, T)×∂Ω.

(1.5)

Lett0∈(0, T] be fixed. Multiplying the equation in (1.5) byv1, and integrating by parts over (0, t0)×(Ω\D1)\G, we obtain

1 2

Z

(Ω\D1)\G

|v1(t0)|²dx=−

t₀

Z

0

Z

(Ω\D1)\G

k(x)∇v1(t)· ∇v1(t)dxdt+

t₀

Z

0

Z

∂((Ω\D1)\G)

k∇v1(t)·nv1(t)dσdt

≤

t₀

Z

0

Z

∂((Ω\D1)\G)

k∇v1(t)·nv1(t)dσdt. (1.6)

Sincevi≡0 on (0, T)×∂Di, from (1.4) and (1.6) we derive Z

(Ω\D1)\G

|v1(t0)|²dx= 0 for allt0∈[0, T].

Hence the unique continuation principle impliesv1≡0 in [0, T)×Ω\D1, that isu1≡cin [0, T)×Ω\D1, where c:=c1=c2. Again, since u1≡0 on (0, T)×∂D1, we derive thatc= 0, that isu1≡0 in [0, T)×Ω\D1. This implies thatf ≡0 on (0, T)×∂Ω, which yields a contradiction. The uniqueness result (a) is then proved.

Concerning the stability result (b), we recall that Vessella [18] proved a continuous dependence of logarithmic type ofD from _∂n^∂ u_|(t0,t1)×Γ (here the interval (t0, t1)⊂[0, T]), in the case where in (1.3)n= 3,k=I3 (I3 is the 3×3 identity matrix), and the temperaturef on (0, T)×∂Ω ismonotonewith respect to the time variablet.

In [8] Canutoet al. have considered the analogous of problem (1.3), but for Neumann boundary conditions (that is the Dirichlet boundary conditionsu= 0 on (0, T)×∂D, u=f on (0, T)×∂Ω appearing in (1.3) are replaced byk∇u·n= 0 on (0, T)×∂D,k∇u·n= 0 on (0, T)×∂D,k∇u·n=gon (0, T)×∂Ω respectively).

They proved a continuous dependence of logarithmic type ofD fromu_|(0,T)×Γ.

The corresponding problem for the elliptic case has been studied too, in a previous paper by Alessandrini et al.[4] who proved a logarithmic stability estimate. Let us point out that, to fix ideas, we have considered in

the present paper the problem of determination of cavities. More generally, we can prove logarithmic stability estimates also when unknown portions of∂Ω are to be determined (see [4, 8] for analogous results). Finally we stress that, in the elliptic case, counterexamples by Alessandrini and Rondi [3] show that logarithmic stability is best possible. This suggests that also in the parabolic case stability estimates better than logarithmic cannot be expected.

We give now a list of oura prioriassumptions on the domains Ω,D, and on the boundary datumf in (1.3), under which we shall prove Theorem 1.1.

We assume that Ω is a bounded domain inRⁿ of class

C^1,1 with constantsR0, E, (1.7)

and thatD is a bounded domain inRⁿ of class

C^1,α, 0< α≤1, with constantsR0, E, (1.8) such that D ⊂ Ω, dist(∂D, ∂Ω) ≥ R0, and Ω\D is connected. For a precise definition of (1.7, 1.8) see Definition 2.1 below. Given M >0, we assume:

|Ω| ≤M Rⁿ₀. (1.9)

Here and in the sequel |Ω|denotes the Lebesgue measure of Ω. We observe that (1.7) and (1.8) imply a lower bound on the diameter of Ω andD respectively. Moreover, by combining (1.7) with (1.9), an upper bound on the diameter of Ω can also be obtained.

We shall assume the following on the Dirichlet datumf:

f ∈H^3/4((0, T), H^1/2(∂Ω)), f 6≡0, and, for a given constantF >0,

kfk1/4,1/2

kfkL²((0,T)×∂Ω)

≤F, (1.10)

where in order to simplify the notations, here and belowkfk1/4,1/2 denotes the normkfkH^1/4((0,T),H^1/2(∂Ω)). We now state the main result of the present paper.

Theorem 1.1. Let Ω be a bounded and connected domain in Rⁿ of class C^1,1, with constants R0, E, and letΓ be a relatively open piece of ∂Ω. Let k(x) be a n×n symmetric matrix-valued function in Ω satisfying assumptions (1.1, 1.2). LetDi, i= {1,2}, be two domains of class C^1,α, 0< α≤ 1, with constants R0, E, such thatDi ⊂Ω, dist(∂Di, ∂Ω)≥R0, and Ω\Di is connected. Let f ∈H^3/4((0, T), H^1/2(∂Ω)) satisfy (1.10) such thatui∈H¹((0, T), H¹(Ω\Di))is solution of (1.3) when D:=Di, and the initial temperatureui0= 0in Ω\Di. If

R0kk∇u1·n−k∇u2·nkL²((0,T)×Γ)≤T¹²R⁽ⁿ₀ ^−1)/2, (1.11) then

d_H(D1, D2)≤CR0

ln T¹²R⁽ⁿ₀ ^−1)/2 kfk1/4,1/2

!

−^κn

, (1.12)

where the constantsC,κdepend onE,α,λ,Λ, ^R_T²⁰,M,F only.

We recall that the Hausdorff distanced_H(D1, D2) between bounded setsD1 andD2 ofRⁿ is the number d_H(D1, D2) := max

sup

x∈D1

dist(x, D2), sup

x∈D2

dist(D1, x)

·

The proof of Theorem 1.1 has the same structure of that in [4] (Ths. 2.1, 2.2) and in [8] (Th. 4.1). As a first step we prove a ln ln-type estimate of the Hausdorff distance between the domains Ω1, Ω2(where Ωi:= Ω\Di), by using as main tools the so-called three spheres and three cylinders inequality for solutions of parabolic equations given in Section 4 (see Ths. 4.1, 4.3, and Cor. 4.2). As a second step, employing in a more refined way the above mentioned inequalities and a geometric lemma (Prop. 5.5), which has been proved in [4], we obtain a logarithmic stability estimate of the Hausdorff distance between Ω1, Ω2, which implies, by a simple reasoning, the desired result,i.e. estimate (1.12). The main difference between the stability result established in [8] (Th. 4.1), and our result, i.e. Theorem 1.1, lies in the hypothesis of regularity of the unknown (a part of the boundary I in [8], and a cavityD in our result), which is of classC^1,1 in [8], and is of classC^1,α, 0< α≤1, in our result.

This difference on the regularity is a consequence of the strong unique continuation principle at the boundary for elliptic operators established by Adolfsson and Escauriaza [1], which need, for the Neumann case, that the boundary of the domain is of classC^1,1, while, for the Dirichlet case, it is sufficient that the boundary is of class C^1,α, 0< α≤1.

The remainder of the paper is organized as follows: in Section 2 we give some notations and definitions;

in Section 3 we introduce the so-called technique of elliptic continuation for solutions of parabolic equations which allow us to define, starting from a solution of a parabolic problem, a solution for a related corresponding elliptic problem. In Section 3 we establish also a Cauchy estimate for the solution of such an elliptic problem.

This estimate will be crucial in Section 4 to prove a three cylinders inequality at the boundary for a parabolic equation. In Section 5 we prove some auxiliary propositions which we shall use in Section 6 to prove Theorem 1.1.

Finally, the appendix (Sect. 7) contains the proof of Lemma 3.3 and some interpolation and traces inequalities, which we use throughout the paper.

2. Notations and definitions

We shall fix the space dimensionn ≥2 throughout the paper. Therefore we shall omit the dependence of the various quantities onn.

We shall use the letterc to denote absolute constants, and the lettersC, ˜C to denote constants depending on some a prioridata. The value of the constants may change from line to line, but we have specified their dependence everywhere they appear.

We shall identifyR² andC.

As usual we shall denote byx= (x1,· · ·, xn) a point in Rⁿ and by x⁰ = (x1,· · ·, xn−1) the first (n−1)- components ofx. X = (y, x) is a point inRⁿ⁺¹, for x∈Rⁿ, whereasX⁰ = (y, x⁰) are the firstn-components ofX.

ByBr(a) (∆r(a), ∆⁰_r(a),Dr(a) respectively) we shall denote the open ball inRⁿ⁺¹(Rⁿ,Rⁿ⁻¹,Crespectively) centered ata, of radiusr. Sometimes we shall write for brevityBr, ∆r, ∆⁰_r,Dr instead ofBr(0), ∆r(0), ∆⁰_r(0), Dr(0), respectively. We shall denote byB_r⁺={X ∈Br s.t. y >0}, ∆⁺_r ={x∈∆r s.t. xn>0}.

When dealing withn+ 1 variables (y, x), we shall denote ∇ =∇^x, div = divx, D² =D²_x. Sometimes we shall write∂_y^kwinstead of ^∂_∂y^kwk,wy instead of ^∂w_∂y andwyyinstead of ^∂_∂y^2w2. Similarly, for brevity, we shall write, for example,kw(y)kL²(Ω)instead ofkw(y,·)kL²(Ω), andR

Ω

|w(y)|²dxinstead ofR

Ω

|w(y, x)|²dx.

When representing locally a boundary as a graph, it will be convenient to use the following notation:

Definition 2.1. Let Ω be a bounded open set inRⁿ. We shall say that a portion Γ of∂Ω is of Lipshitz class (resp. of classC^1,α, 0< α≤1) with constantsR0,E >0, if, for anyP∈Γ, there exists a rigid transformation

of coordinates under which we haveP= 0 and

Ω∩∆R0={x∈∆⁰_R0 s.t. xn > ϕ(x⁰)}, whereϕis aC^0,1 function (resp. ϕis aC^1,α function) on ∆⁰_R0 ⊂Rⁿ⁻¹ satisfying

ϕ(0) = 0 (and resp. ϕ(0) =|∇ϕ(0)|= 0) and

kϕkC^0,1(∆⁰_R

0)≤ER0 (resp. kϕkC^1,α(∆⁰_R

0)≤ER0).

Remark 2.2. We have chosen to normalize all norms in such a way that their terms are dimensionally homo- geneous, and coincide with the standard definition whenR0 = 1 andT = 1. For instance, the norm appearing above is meant as follows

kϕk^C1,α(∆0_R0⁾:=kϕkL^∞(∆⁰_R

0)+R0k∇ϕkL^∞(∆⁰_R

0)+R^1+α₀ [∇ϕ]_α,∆⁰

R0

, where

[∇ϕ]_α,∆0 R0

:= sup

x,y∈∆0 R0 x6=y

|∇ϕ(x)− ∇ϕ(y)|

|x−y|^α ,

and|·|is the Euclidean norm. Similarly we shall set

kuk^C0,1((0,T)×Ω):=kuk^L∞((0,T)×Ω)+R0[u]_1,(0,T)×Ω, where

[u]_1,(0,T)×Ω:= sup

(t,x),(s,y)∈(0,T)×Ω (t,x)6=(s,y)

|u(t, x)−u(s, y)|

|(t, x)−(s, y)| ,

kuk²H¹((0,T),H¹(Ω)) :=

ZT 0

Z

Ω

(|u|²+T²|ut|²+R²₀|∇u|²)dxdt,

and so on for boundary and trace norms such ask·kL²((0,T)×Ω),k · kH^1/4((0,T),H^1/2(∂Ω)).

3. Elliptic continuation for solutions of parabolic equations

In this section we introduce the so-called technique of elliptic continuation for solutions of parabolic equations (see Landis and Oleinik [14] or Lin [15]), which can be traced back to the pioneering work by Ito and Yamabe [12], who introduced this technique in 1959 to prove unique continuation properties for solutions of

∂tu−div(k(x)∇u) = 0 in (0, T)×Ω⁰. (3.1) Roughly speaking this technique consists in the following idea: fixing t0 ∈ (0, T), a solution of the parabolic equation (3.1) can be continued to a function w(t0;y, x) (for values of y in an appropriate interval) which

satisfies an elliptic equation iny,x(see Prop. 3.1 below). In this way many properties of the solutions of elliptic equations can be transferred to solutions of parabolic equations.

Here and below we assume that Ω⁰ is a bounded domain inRⁿ,n≥2, of classC^1,α, 0< α≤1, with constant R0,E,x0∈∂Ω⁰,R∈(0, R0/2], andt0∈(0, T). Moreover we suppose thatkis an×nsymmetric matrix-valued function in Ω⁰ satisfying assumptions (1.1, 1.2) (with Ω replaced by Ω⁰), andu∈H¹((0, T), H¹(Ω⁰∩∆2R(x0))) is a nonidentically zero solution of

(ut−div(k(x)∇u) = 0 in (0, T)×(Ω⁰∩∆2R(x0)),

u= 0 on (0, T)×((∂Ω⁰)∩∆2R(x0)). (3.2) The main result in this section is the following:

Proposition 3.1. LetA:= min√ 2δ, t0√

aR , whereδ:= _8eπλ^R ,aR:=_λcP¹_R2, andcP is the Poincar´e constant.

There exists a functionw∈C^ω((−A, A), H¹(Ω⁰ ∩∆R/2(x0))) solution of the following problem















wyy+ div(k(x)∇w) = 0 in (−A, A)×(Ω⁰∩∆R/2(x0)), w(0) =u(t0) in Ω⁰∩∆R/2(x0),

wy(0) = 0 in Ω⁰∩∆R/2(x0),

w = 0 on (−A, A)×((∂Ω⁰)∩∆R/2(x0)).

(3.3)

Moreover, for

r≤ 3A

32eπλ, (3.4)

and

ρ:= 8 3

√2eπλr, ρe:= 2√ 2ρ, the following inequality holds:

Z

(R×Ω⁰)∩Br(X0)

|w|²dX ≤Cr



 Z

Ω⁰∩∆_ρ/4(x0)

|u(t0)|²dx





β

1 r

Z

(R×Ω⁰)∩Bρe(X0)

|w|²dX





1−β

, (3.5)

whereX0∈Rⁿ⁺¹is the point(0, x0), the constantC≥1depends onλonly, andβ:= _1+α^αβ ,β∈(0,1)depending onλonly.

(We observe that the choice ofrin (3.4) implies that ρ < A.) We recall thate C^ω(R, Z) denotes the space of real analytic variable functions with values in a Banach spaceZ, and dX (resp. dx) is the (n+ 1)-dimensional (resp. n-dimensional) volume Lebesgue measure.

We precede the proof of Proposition 3.1 by some preliminary lemmas.

Lemma 3.2. Under the assumptions of Proposition 3.1, letη∈C²[0,+∞)be a cut-off function satisfying:

η(t) =

(1 fort∈[0, t0]

0for t∈[T,+∞) , and |η⁰| ≤ c T−t0·

There exists a unique solutionu1∈C((0, T), H¹(Ω⁰ ∩∆2R(x0)))∩C¹([0, T), L²(Ω⁰ ∩∆2R(x0)))of the problem:









u1t−div(k(x)∇u1) = 0 in (0,+∞)×(Ω⁰∩∆2R(x0)), u1(0) = 0 in Ω⁰∩∆2R(x0),

u1=g on (0,+∞)×∂(Ω⁰∩∆2R(x0)),

(3.6)

whereg:=η(t)u. Moreover, for all t≥0, we have

ku1(t)kH¹(Ω⁰∩∆2R(x0))≤ce^{−aR(t−T)+}C1H, (3.7) where(t−T)+:= max(0,(t−T)),

C1:=

λ T

T−t0

e^T−tT0 + R²₀ T−t0

¹₂

, (3.8)

and

H := max

0≤t≤Tku(t)kH¹(Ω⁰∩∆2R(x0)). (η⁰ denotes the derivative ofη, andaR is as in Prop. 3.1.)

Proof of Lemma 3.2. The proof follows step by step, up to the obvious changes, from the proof of Lemma 3.1.2

in [8].

Let us still denote by u1 the extension by 0 of u1 to R×(Ω⁰∩∆2R(x0)), and let fu1(µ, x) be the Fourier transform ofu1(t, x) with respect to the time variablet, that is

f

u1(µ, x) := 1 2π

+∞

Z

−∞

e^−iµtu1(t, x)dt. (3.9)

The following result holds:

Lemma 3.3. Under the assumptions of Proposition 3.1, let fu1(µ, x) be as above. Then fu1 ∈ C^ω(R, H¹(Ω⁰∩

∆2R(x0))) solves

(iµfu1−div(k(x)∇fu1) = 0 in R×(Ω⁰∩∆2R(x0)), f

u1 = 0 on R×((∂Ω⁰)∩∆2R(x0)).

Moreover

kfu1(µ)kH¹(Ω⁰∩∆_R/2(x₀))≤cC1He⁻√

|µ|δ

T+ 1

aR/4

· (3.10)

(Here the constants c,C1,H are as in Lem. 3.2, andδ:= _8eπλ^R is as in Prop. 3.1.)

Proof of Lemma 3.3. See the appendix, Section 7.

Lemma 3.4. Under the assumptions of Proposition 3.1, let` >0, and letρ∈(0, R). Forf ∈H<sup>1</sup>(Ω<sup>0</sup>∩∆2ρ(x0)), assume thatw∈C<sup>ω</sup>((−2`,2`), H¹(Ω⁰∩∆2ρ(x0))solves















wyy+ div(k(x)∇w) = 0 in (−2`,2`)×(Ω⁰∩∆2ρ(x0)), w(0) =f in Ω⁰∩∆2ρ(x0),

wy(0) = 0 in Ω⁰∩∆2ρ(x0),

w= 0 on (−2`,2`)×((∂Ω⁰)∩∆2ρ(x0)).

Then, for

ρ1 = πeλ 1

ρ² + 1

`²

1/2!₋1

, (3.11)

ρ2 = min

ρ1, ρ 4√ λ

, (3.12)

ρ3 = 1 2(ρ−√

λρ2), (3.13)

and for every y∈ −³8ρ1,³₈ρ1

, the following inequality holds:

Z

Ω⁰∩∆_ρ3(x₀)

|wy(y)|²+|∇w(y)|²

dx≤C

k∇fk²_L2(Ω⁰∩∆_ρ/4(x0))

β

× 1

`ρ<sup>2</sup>kwk<sup>2</sup><sub>L</sub>2((−2`,2`)×(Ω⁰∩∆2ρ(x0)))+k∇fk²¹_L2⁻(Ω^β0∩∆_ρ/4(x₀))

1−β

, (3.14) where the constant C depends onλand `ρ⁻¹ only, andβ is as in Proposition 3.1.

Proof of Lemma 3.4. We divide the proof into three steps.

Step 1: In this step we prove that the power series

+∞

X

j=0

∂y^jw(0)z^j

j! (3.15)

converges in C^1,α(Ω⁰∩∆¹

4ρ(x0))∩H_loc² (Ω⁰∩∆³

4ρ(x0)) for every complex numberz such that|z|< ρ1. Let us denoteQ0:= (−2`,2`)×(Ω⁰∩∆2ρ(x0)),Q1:= (−`, `)×(Ω⁰∩∆ρ(x0)). By a slight modification of the arguments used to prove Lemma 3.3, we obtain

k∂_y^jwk²L²(Q₁)≤(C2j²)^jkwk²L²(Q₀) for everyj≥1, (3.16) where

C2=π²λ² 1

ρ² + 1

`²

· (3.17)

Let us fixj≥1 and let us denote

U(y, x) =∂_y^jw(y, x) in (−`, `)×(Ω⁰∩∆ρ(x0)). (3.18)

We have thatU ∈C^ω((−`, `), H¹(Ω⁰∩∆ρ(x0))) solves

(∂_yy² U+ div(k(x)∇U) = 0 in (−`, `)×(Ω⁰∩∆ρ(x0)),

U = 0 on (−`, `)×((∂Ω⁰)∩∆ρ(x0)). (3.19) By standardC^1,α estimates (see Gilbarg and Trudinger [10]) we have

kUkC^1,α((−^{`</sup>4,<sup>`}₄)×(Ω⁰∩∆ρ

4(x0))≤ C

ρⁿ⁺¹² kUkL²((−`,`)×(Ω⁰∩∆ρ(x0)), (3.20) where the constant C depends onE, α, λ, Λ, R0/ρ. From (3.16, 3.18, 3.20) we obtain, for everyy ∈(−^{`</sup>4,<sup>`}₄), and for everyj≥1,

∂_y^jw(y)

C^1,α(Ω⁰∩∆ρ

4(x0))≤ C ρⁿ⁺¹² C

j 2

2j^jkwkL²(Q0). (3.21) So (3.21) yields the convergence in C^1,α(Ω⁰∩∆¹

4ρ(x0)) of the power series (3.15) in the diskDρ₁, whereρ1 is given by (3.11).

For anyϕ∈L²(Ω⁰∩∆ρ(x0)), let

F(y) :=

Z

Ω⁰∩∆_ρ(x₀)

w(y)ϕdx.

By (3.16) and by the interpolation inequality (7.10) (see the Appendix) we obtain, for everyj≥1,

|F^(j)(y)|²≤ C

`C<sub>2</sub><sup>j</sup>(j+ 1)<sup>2(j+1)</sup>kwk<sup>2</sup>L<sup>2</sup>(Q0)kϕk<sup>2</sup>L<sup>2</sup>(Ω<sup>0</sup>∩∆<sub>ρ</sub>(x<sub>0</sub>)) (3.22) for everyy∈(−`, `), whereCdepends onλand`ρ⁻¹ only. Therefore, for everyj ≥1,

Z

Ω⁰∩∆_ρ(x₀)

|∂y^jw(y)|²dx≤C

`C<sub>2</sub><sup>j</sup>(j+ 1)<sup>2(j+1)</sup>kwk<sup>2</sup>L<sup>2</sup>(Q0) (3.23) for everyy∈(−`, `). Let us fixj≥1 andy∈(−`, `), and let us denote

g(y) =∂_y^j+2w(y) in Ω⁰∩∆ρ(x0), (3.24)

U(y) =∂_y^jw(y) in Ω⁰∩∆ρ(x0). (3.25)

We have thatU(y)∈H¹(Ω⁰∩∆ρ(x0)) solves

(div(k∇U(y)) =−g(y) in Ω⁰∩∆ρ(x0),

U(y) = 0 on (∂Ω⁰)∩∆ρ(x0). (3.26)

From Caccioppoli inequality we have k∇U(y)k²_L2(Ω⁰∩∆3

4ρ(x0))≤C

ρ²kg(y)k²_L2(Ω⁰∩∆ρ(x0))+ 1

ρ²kU(y)k²_L2(Ω⁰∩∆ρ(x0))

, (3.27)

whereCdepends onλonly. Now letx∈Ω⁰∩∆³

4ρ(x0) and let rbe such that

∆r(x)⊂Ω⁰∩∆³

4ρ(x0).

Choosing as test functionsV(y) = (η²Ux_i(y))x_i,i= 1, ..., n, whereηis a cut off function, we obtain, by standard H_loc² estimates [10], and by (3.27)

kD²U(y)k²L²(∆r

2(x))≤C 1

r² +Λ²

R²₀ ρ²kg(y)k²_L2(Ω⁰∩∆_ρ(x₀))+ 1

ρ²kU(y)k²_L2(Ω⁰∩∆ρ(x0))

, (3.28)

whereCdepends onλonly. By (3.23–3.25, 3.27, 3.28) we have, for everyj≥1, Z

∆r 2(x)

|∇∂_y^jw(y)|²dx≤ C

`ρ²C₂^j(j+ 3)^2(j+3)kwk²L²(Q₀), (3.29) Z

∆r 2(x)

|D²∂y^jw(y)|²dx≤ C

`ρ² 1

r² +Λ² R²₀

C₂^j(j+ 3)^2(j+3)kwk²L²(Q0), (3.30)

where the constantC in (3.29, 3.30) depends onλand`ρ⁻¹only. Finally (3.29, 3.30) yield the convergence in H_loc² (Ω⁰∩∆³

4ρ(x0)) of the power series (3.15) in the diskDρ1, whereρ1 is given by (3.11).

Let us denote, forx∈Ω⁰∩∆¹

4ρ(x0), W(z, x) : =

+∞

X

j=0

∂y^jw(0, x)z^j

j!, forz∈Dρ₁, v(ξ, x) : =W(iξ, x), for|ξ|< ρ1.

Step 2: In this step we prove that for everyξ∈(−ρ2, ρ2) (ρ2 as in (3.12)) we have Z

Ω⁰∩∆_ρ(ξ)(x₀)

(|vξ(ξ)|²+k∇v(ξ)· ∇v(ξ))dx≤ Z

Ω⁰∩∆ρ 4

(x₀)

|∇f|²dx, (3.31)

where

ρ(ξ) = ρ 4−√

λ|ξ|. (3.32)

First, let us observe thatvis real and solves the following hyperbolic initial boundary value problem:















vξξ(ξ)−div(k∇v(ξ)) = 0 in (−ρ1, ρ1)×(Ω⁰∩∆^ρ

4(x0)), v(0, x) =f(x) in Ω⁰∩∆^ρ₄(x0),

vξ(0, x) = 0 in Ω⁰∩∆^ρ

4(x0),

v = 0 on (−ρ1, ρ1)×((∂Ω⁰)∩∆^ρ₄(x0)).

(3.33)

We shall derive estimate (3.31) from an energy estimate for the problem (3.33). To this aim, let us denote E(ξ) =1

2 Z

Ω⁰∩∆_ρ(ξ)(x₀)

(|vξ(ξ)|²+k∇v(ξ)· ∇v(ξ))dx. (3.34)

Sinceξ→v(ξ) is an analytic function from (−ρ1, ρ1) to C^1,α(Ω⁰∩∆¹

4ρ(x0)) we have that∂_ξ^jv(ξ)∈C^1,α(Ω⁰∩

∂∆ρ(ξ)(x0)) for everyξ∈(−ρ2, ρ2) and for everyj≥1, whereρ2 is given by (3.12). For everyξ∈(−ρ2, ρ2), by the coarea formula we have the following equality

E(ξ) = 1 2

ρ(ξ)Z

0

dη Z

Ω⁰∩∂∆η(x0)

(|vξ(ξ)|²+k∇v(ξ)· ∇v(ξ))dσ, (3.35)

where dσis the (n−1)-dimensional surface Lebesgue measure. The derivative ofE(ξ) is equal to

E⁰(ξ) =

ρ(ξ)Z

0

dη Z

Ω⁰∩∂∆_η(x₀)

(vξ(ξ)vξξ(ξ) +k∇v(ξ)· ∇vξ(ξ))dσ

−

√λ 2

Z

Ω⁰∩∂∆_ρ(ξ)(x₀)

(|vξ(ξ)|²+k∇v(ξ)· ∇v(ξ))dσ

= Z

Ω⁰∩∆_ρ(ξ)(x₀)

(vξ(ξ)vξξ(ξ) +k∇v(ξ)· ∇vξ(ξ))dx

−

√λ 2

Z

Ω⁰∩∂∆_ρ(ξ)(x0)

(|vξ(ξ)|²+k∇v(ξ)· ∇v(ξ))dσ. (3.36)

Moreover, sincev(ξ)∈H_loc² (Ω⁰∩∆³

4ρ(x0)), a simple calculation gives

k∇v(ξ)· ∇vξ(ξ) =−div(k∇v(ξ))vξ+ div(kvξ∇v(ξ)) in Ω⁰∩∆ρ(ξ)(x0). (3.37) So by Green’s formula and the fact thatv= 0 on (−ρ1, ρ1)×((∂Ω⁰)∩∆^ρ

4(x0)), from (3.36, 3.37) we obtain E⁰(ξ) =

Z

Ω⁰∩∂∆_ρ(ξ)(x0)

k∇v(ξ)·nvξ(ξ)dσ−

√λ 2

Z

Ω⁰∩∂∆_ρ(ξ)(x0)

(|vξ(ξ)|²+k∇v(ξ)· ∇v(ξ))dσ,

wherendenotes the outer unit normal to Ω⁰∩∂∆ρ(ξ)(x0). We have

|k∇v(ξ)·nvξ(ξ)| ≤ (k∇v(ξ)· ∇v(ξ))^1/2(kn·n)^1/2|vξ(ξ)|

≤

√λ

2 (|vξ(ξ)|²+k∇v(ξ)· ∇v(ξ)).

ThereforeE⁰(ξ)≤0, hence the function Eis decreasing, so that E(ξ)≤E(0) and (3.31) follows.

Step 3: In this step we prove the assertion of Lemma 3.4. For everyz∈Dρ₁ let us set G(z) :=

Z

Ω⁰∩∆_ρ3(x₀)

(Wz(z)²+k∇W(z)· ∇W(z))dx (3.38)

(whereρ3 is defined in (3.13)), and let ²=

Z

Ω⁰∩∆ρ 4

(x₀)

k∇f · ∇fdx. (3.39)

Letρ⁰1∈(0, ρ1). By (3.23) and (3.27) we obtain

|G(z)| ≤ C

`ρ<sup>2</sup>(1−ρ<sup>0</sup><sub>1</sub>ρ<sup>−</sup><sub>1</sub><sup>1</sup>)<sup>8</sup>kwk<sup>2</sup>L<sup>2</sup>(Q0), for everyz∈Dρ<sup>0</sup><sub>1</sub>, (3.40) whereCdepends onλand`ρ⁻¹ only. On the other side (3.31) gives

|G(iξ)| ≤², for everyξ∈(−ρ2, ρ2). (3.41) From (3.40, 3.41) and the analytic continuation estimate (see Isakov [11]) we obtain

G(y) = Z

Ω⁰∩∆_ρ3(x₀)

(wy(y)²+k∇w(y)· ∇w(y))dx≤ 1 (1−ρ⁰₁ρ⁻₁¹)⁸





 Z

Ω⁰∩∆ρ 4

(x₀)

k∇f· ∇fdx







ω(0,y)

×





 C

`ρ²kwk²L²(Q0)+ Z

Ω⁰∩∆ρ 4

(x₀)

k∇f· ∇fdx







1−ω(0,y)

, (3.42)

whereω(ξ, y) is the harmonic measure of{iξs.t. ξ∈[−^ρ2²,^ρ₂²]}with respect to{y+iξ∈Cs.t. y²+ξ²= (ρ⁰1)²} and C depends on λ and `ρ⁻¹ only. Now, let us choose ρ⁰₁ = ³₄ρ1, so that ^ρ₂² < ρ⁰₁ < ρ1. We have that ω(0, y)≥β >0 for everyy∈(−8³ρ1,³₈ρ1), whereβ depends onλand Λ only. Therefore estimate (3.14) follows by (3.42).

The proof of Lemma 3.4 is complete.

We are now in a position to prove Proposition 3.1.

Proof of Proposition 3.1. Let us define

w1(y, x) := 1 2π

+∞

Z

−∞

e^it0µfu1(µ, x) cosh(p

−iµy)dµ,

wherefu1has been introduced in (3.9). By (3.10) it follows thatw1(y)∈H¹(Ω⁰∩∆R/2(x0)), fory∈(−√ 2δ,√

2δ).

Moreoverw1∈C^ω((−√ 2δ,√

2δ), H¹(Ω⁰∩∆R/2(x0))) and solves















w1yy+ div(k(x)∇w1) = 0 in (−√ 2δ,√

2δ)×(Ω⁰∩∆R/2(x0)), w1(0) =u1(t0) in Ω⁰∩∆R/2(x0),

w1y(0) = 0 in Ω⁰∩∆R/2(x0), w1 = 0 on (−√

2δ,√

2δ)×((∂Ω⁰)∩∆R/2(x0)).