New global stability estimates for the Calderón problem in two dimensions
aa r X i v : . [ m a t h . A P ] M a r NEW GLOBAL STABILITY ESTIMATES FOR THECALDERÓN PROBLEM IN TWO DIMENSIONS
MATTEO SANTACESARIA
Abstract.
We prove a new global stability estimate for the Gel’fand-Calderón inverse problem on a two-dimensional bounded domain. Specif-ically, the inverse boundary value problem for the equation − ∆ ψ + v ψ =0 on D is analysed, where v is a smooth real-valued potential of conduc-tivity type defined on a bounded planar domain D . The main feature ofthis estimate is that it shows that the more a potential is smooth, themore its reconstruction is stable. Furthermore, the stability is proven todepend exponentially on the smoothness, in a sense to be made precise.The same techniques yield a similar estimate for the Calderón problemfor the electrical impedance tomography. Introduction
Let D ⊂ R be a bounded domain equipped with a potential given by afunction v ∈ L ∞ ( D ) . The corresponding Dirichlet-to-Neumann map is theoperator Φ : H / ( ∂D ) → H − / ( ∂D ) , defined by(1.1) Φ( f ) = ∂u∂ν (cid:12)(cid:12)(cid:12)(cid:12) ∂D , where f ∈ H / ( ∂D ) , ν is the outer normal of ∂D , and u is the H ( D ) -solution of the Dirichlet problem(1.2) ( − ∆ + v ) u = 0 on D, u | ∂D = f. Here we have assumed that(1.3) is not a Dirichlet eigenvalue for the operator − ∆ + v in D. The following inverse boundary value problem arises from this construc-tion:
Problem 1.
Given Φ , find v on D . This problem can be considered as the Gel’fand inverse boundary valueproblem for the Schrödinger equation at zero energy (see [10], [17]) as well
Mathematics Subject Classification.
Key words and phrases.
Calderón problem, electrical impedance tomography, Schrödingerequation, global stability in 2D, generalised analytic functions. as a generalization of the Calderón problem for the electrical impedancetomography (see [7], [17]), in two dimensions.It is convenient to recall how the above problem generalises the inverseconductivity problem proposed by Calderón. In the latter, D is a bodyequipped with an isotropic conductivity σ ( x ) ∈ L ∞ ( D ) (with σ ≥ σ min > ), v ( x ) = ∆ σ / ( x ) σ / ( x ) , x ∈ D, (1.4) Φ = σ − / Λ σ − / + ∂σ / ∂ν ! , (1.5)where σ − / , ∂σ / /∂ν in (1.5) denote the multiplication operators by thefunctions σ − / | ∂D , ∂σ / /∂ν | ∂D , respectively and Λ is the voltage-to-currentmap on ∂D , defined as(1.6) Λ f = σ ∂u∂ν (cid:12)(cid:12)(cid:12)(cid:12) ∂D , where f ∈ H / ( ∂D ) , ν is the outer normal of ∂D , and u is the H ( D ) -solution of the Dirichlet problem(1.7) div( σ ∇ u ) = 0 on D, u | ∂D = f. Indeed, the substitution u = ˜ uσ − / in (1.7) yields ( − ∆ + v )˜ u = 0 in D with v given by (1.4). The following problem is called the Calderón problem: Problem 2.
Given Λ , find σ on D. We remark that Problems 1 and 2 are not overdetermined, in the sense thatwe consider the reconstruction of a real-valued function of two variables fromreal-valued inverse problem data dependent on two variables. In addition,the history of inverse problems for the two-dimensional Schrödinger equationat fixed energy goes back to [8].There are several questions to be answered in these inverse problems: toprove the uniqueness of their solutions (e.g. the injectivity of the map v → Φ for Problem 1), the reconstruction and the stability of the inverse map.In this paper we study interior stability estimates for the two problems.Let us consider, for instance, Problem 1 with a potential of conductivitytype. We want to prove that given two Dirichlet-to-Neumann operators,respectively Φ and Φ , corresponding to potentials, respectively v and v on D , we have that k v − v k L ∞ ( D ) ≤ ω ( k Φ − Φ k H / → H − / ) , EW GLOBAL STABILITY IN 2D 3 where the function ω ( t ) → as fast as possible as t → . For Problem 2similar estimates are considered.There is a wide literature on the Gel’fand-Calderón inverse problem. In thecase of complex-valued potentials the global injectivity of the map v → Φ was firstly proved in [17] for D ⊂ R d with d ≥ and in [6] for d = 2 with v ∈ L p : in particular, these results were obtained by the use of globalreconstructions developed in the same papers. A global stability estimatefor Problem 1 and 2 for d ≥ was first found by Alessandrini in [1]; thisresult was recently improved in [21]. In the two-dimensional case the firstglobal stability estimate for Problem 1 was given in [23].Global results for Problem 2 in the two dimensional case have been foundmuch earlier than for Problem 1. In particular, global uniqueness was firstproved in [16] for conductivities in the W ,p ( D ) class ( p > ) and afterin [2] for L ∞ conductivities. The first global stability result was given in[14], where a logarithmic estimate is obtained for conductivities with twocontinuous derivatives. This result was improved in [4], where the same kindof estimate is obtained for Hölder continuous conductivities.The research line delineated above is devoted to prove stability estimatesfor the least regular potentials/conductivities possible. Here, instead, wefocus on the opposite situation, i.e. smooth potentials/conductivities, andtry to answer another question: how the stability estimates vary with respectto the smoothness of the potentials/conductivities.The results, detailed below, also constitute a progress for the case of non-smooth potentials: they indicate stability dependence of the smooth part ofa singular potential with respect to boundary value data.We will assume for simplicity that D is an open bounded domain in R , ∂D ∈ C ,v ∈ W m, ( R ) for some m > , supp v ⊂ D, (1.8)where W m, ( R ) = { v : ∂ J v ∈ L ( R ) , | J | ≤ m } , m ∈ N ∪ { } , (1.9) J ∈ ( N ∪ { } ) , | J | = J + J , ∂ J v ( x ) = ∂ | J | v ( x ) ∂x J ∂x J . Let k v k m, = max | J |≤ m k ∂ J v k L ( R ) . MATTEO SANTACESARIA
The last (strong) hypothesis is that we will consider only potentials of con-ductivity type, i.e.(1.10) v = ∆ σ / σ / , for some σ ∈ L ∞ ( D ) , with σ ≥ σ min > . The main results are the following.
Theorem 1.1.
Let the conditions (1.3) , (1.8) , (1.10) hold for the potentials v , v , where D is fixed, and let Φ , Φ be the corresponding Dirichlet-to-Neumann operators. Let k v j k m, ≤ N , j = 1 , , for some N > . Thenthere exists a constant C = C ( D, N, m ) such that (1.11) k v − v k L ∞ ( D ) ≤ C (log(3 + k Φ − Φ k − )) − α , where α = m − and k Φ − Φ k = k Φ − Φ k H / → H − / . Theorem 1.2.
Let σ , σ be two isotropic conductivities such that ∆( σ / j ) /σ / j satisfies conditions (1.8) , where D is fixed and < σ min ≤ σ j ≤ σ max < + ∞ for j = 1 , and some constants σ min and σ max . Let Λ , Λ be the cor-responding Dirichlet-to-Neumann operators and k ∆( σ / j ) /σ / j k m, ≤ N , j = 1 , , for some N > . We suppose, for simplicity, that supp ( σ j − ⊂ D for j = 1 , . Then, for any α < m there exists a constant C = C ( D, N, σ min , σ max , m, α ) such that (1.12) k σ − σ k L ∞ ( D ) ≤ C (log(3 + k Λ − Λ k − )) − α , where k Λ − Λ k = k Λ − Λ k H / → H − / . The main feature of these estimates is that, as m → + ∞ , we have α → + ∞ . In addition we would like to mention that, under the assumptionsof Theorems 1.1 and 1.2, according to instability estimates of Mandache [15]and Isaev [13], our results are almost optimal. Note that, in the linear ap-proximation near the zero potential, Theorem 1.1 (without condition (1.10))was proved in [22]. In dimension d ≥ a global stability estimate similar toour result (with respect to dependence on smoothness) was proved in [21].The proof of Theorem 1.1 relies on the ¯ ∂ -techniques introduced by Beals–Coifman [5], Henkin–R. Novikov [12], Grinevich–S. Novikov [11] and devel-oped by R. Novikov [17] and Nachman [16] for solving the Calderón problemin two dimensions.The Novikov–Nachman method starts with the construction of a specialfamily of solutions ψ ( x, λ ) of equation (1.2), which was originally introducedby Faddeev in [9]. These solutions have an exponential behaviour dependingon the complex parameter λ and they are constructed via some function µ ( x, λ ) (see (2.5)). One of the most important property of µ ( x, λ ) is that it EW GLOBAL STABILITY IN 2D 5 satisfies a ¯ ∂ -equation with respect to the variable λ (see equation (2.8)), inwhich appears the so-called Faddeev generalized scattering amplitude h ( λ ) (defined in (2.6)). On the contrary, if one knows h ( λ ) for every λ ∈ C ,it is possible to recover µ ( x, λ ) via this ¯ ∂ -equation. Starting from thesearguments we will prove that the map h ( λ ) → µ ( z, λ ) satisfies an Höldercondition, uniformly in the space variable z . This is done in Section 4.Another part of the method relates the scattering amplitude h ( λ ) to theDirichlet-to-Neumann operator Φ . In the present paper this is done usingthe Alessandrini identity (see [1]) and an estimate of h ( λ ) for high values of | λ | given in [19]. We find that the map Φ → h has logarithmic stability insome natural norm (Proposition 3.3). This is explained in Section 3.The final part of the method for the two problems is quite different. ForProblem 2, in order to recover σ ( x ) from µ ( x, λ ) , we use a limit found for thefirst time in [16]. Instead, for Problem 1, we use an explicit formula for v ( x ) which involves the scattering amplitude h ( λ ) , µ ( x, λ ) and its first (complex)derivative with respect to z = x + ix (see formula (5.3)). The two resultsare presented in section 5 and yield the proofs of Theorems 1.1 and 1.2.This work was fulfilled in the framework of researches under the directionof R. G. Novikov. 2. Preliminaries
In this section we recall some definitions and properties of the Faddeevfunctions, the above-mentioned family of solutions of equation (1.2), whichwill be used throughout all the paper.Following [16], we fix some < p < and define ψ ( x, k ) to be the solution(when it exists unique) of(2.1) ( − ∆ + v ) ψ ( x, k ) = 0 in R , with e − ixk ψ ( x, k ) − ∈ W , ˜ p ( R ) = { u : ∂ J u ∈ L ˜ p ( R ) , | J | ≤ } , where x = ( x , x ) ∈ R , k = ( k , k ) ∈ V ⊂ C , V = { k ∈ C : k = k + k = 0 } (2.2)and(2.3) p = 1 p − . The variety V can be written as { ( λ, iλ ) : λ ∈ C } ∪ { ( λ, − iλ ) : λ ∈ C } . Wehenceforth denote ψ ( x, ( λ, iλ )) by ψ ( x, λ ) and observe that, since v is real-valued, uniqueness for (2.1) yields ψ ( x, ( − ¯ λ, i ¯ λ )) = ψ ( x, ( λ, iλ )) = ψ ( x, λ ) so that, for reconstruction and stability purpose, it is sufficient to work onthe sheet k = ( λ, iλ ) . MATTEO SANTACESARIA
We now identify R with C and use the coordinates z = x + ix , ¯ z = x − ix , ∂∂z = 12 (cid:18) ∂∂x − i ∂∂x (cid:19) , ∂∂ ¯ z = 12 (cid:18) ∂∂x + i ∂∂x (cid:19) , where ( x , x ) ∈ R .Then we define ψ ( z, λ ) = ψ ( x, λ ) , (2.4) µ ( z, λ ) = e − izλ ψ ( z, λ ) , (2.5) h ( λ ) = Z D e i ¯ z ¯ λ v ( z ) ψ ( z, λ ) d Re z d Im z, (2.6)for z, λ ∈ C .Throughout all the paper c ( α, β, . . . ) is a positive constant depending onparameters α, β, . . . We now restate some fundamental results about Faddeev functions. Inthe following statement ψ denotes σ / . Proposition 2.1 (see [16]) . Let D ⊂ R be an open bounded domain with C boundary, v ∈ L p ( R ) , < p < , supp v ⊂ D , k v k L p ( R ) ≤ N , be such thatthere exists a real-valued ψ ∈ L ∞ ( R ) with v = (∆ ψ ) /ψ , ψ ( x ) ≥ c > and ψ ≡ outside D . Then, for any λ ∈ C there is a unique solution ψ ( z, λ ) of (2.1) with e − izλ ψ ( · , λ ) − in L ˜ p ∩ L ∞ ( ˜ p is defined in (2.3) ).Furthermore, e − izλ ψ ( · , λ ) − ∈ W , ˜ p ( R ) and (2.7) k e − izλ ψ ( · , λ ) − k W s, ˜ p ≤ c ( p, s ) N | λ | s − , for ≤ s ≤ and λ sufficiently large.The function µ ( z, λ ) defined in (2.5) satisfies the equation (2.8) ∂µ ( z, λ ) ∂ ¯ λ = 14 π ¯ λ h ( λ ) e − λ ( z ) µ ( z, λ ) , z, λ ∈ C , in the W , ˜ p topology, where h ( λ ) is defined in (2.6) and the function e − λ ( z ) is defined as follows: (2.9) e λ ( z ) = e i ( zλ +¯ z ¯ λ ) . In addition, the functions h ( λ ) and µ ( z, λ ) satisfy (cid:13)(cid:13)(cid:13)(cid:13) h ( λ )¯ λ (cid:13)(cid:13)(cid:13)(cid:13) L r ( R ) ≤ c ( r, N ) , for all r ∈ (˜ p ′ , ˜ p ) , p + 1˜ p ′ = 1 , (2.10) sup z ∈ C k µ ( z, · ) − k L r ( C ) ≤ c ( r, D, N ) , for all r ∈ ( p ′ , ∞ ] (2.11) EW GLOBAL STABILITY IN 2D 7 and | h ( λ ) | ≤ c ( p, D, N ) | λ | ε , (2.12) k µ ( · , λ ) − ψ k W , ˜ p ≤ c ( p, D, N ) | λ | ε , (2.13) for λ ≤ λ ( p, D, N ) and < ε < p ′ , where p + p ′ = 1 . Remark.
Equation (2.8) means that µ is a generalised analytic function in λ ∈ C (see [24]). In two-dimensional inverse scattering for the Schrödingerequation, the theory of generalised analytic functions was used for the firsttime in [11].We recall that if v ∈ W m, ( R ) with supp v ⊂ D , then k ˆ v k m < + ∞ , where ˆ v ( p ) = (2 π ) − Z R e ipx v ( x ) dx, p ∈ C , (2.14) k u k m = sup p ∈ R | (1 + | p | ) m/ u ( p ) | , (2.15)for a test function u .In addition, if v ∈ W m, ( R ) with supp v ⊂ D and m > , we have, bySobolev embedding, that(2.16) k v k L ∞ ( D ) ≤ c ( D ) k v k m, , so, in particular, the hypothesis v ∈ L p ( R ) , supp v ⊂ D , in the statementof Proposition 2.1 is satisfied for every < p < (since D is bounded).The following lemma is a variation of a result in [19]: Lemma 2.2.
Under the assumption (1.8) , there exists R = R ( m, k ˆ v k m ) > such that (2.17) | h ( λ ) | ≤ π k ˆ v k m (1 + 4 | λ | ) − m/ , for | λ | > R. Proof.
We consider the function H ( k, p ) defined as(2.18) H ( k, p ) = 1(2 π ) Z R e i ( p − k ) x v ( x ) ψ ( x, k ) dx, for k ∈ V (where V is defined in (2.2)), p ∈ R and ψ ( x, k ) as defined at thebeginning of this section.We deduce that h ( λ ) = (2 π ) H ( k ( λ ) , k ( λ ) + k ( λ )) , for k ( λ ) = ( λ, iλ ) . By[19, Corollary 1.1] we have(2.19) | H ( k, p ) | ≤ k ˆ v k m (1 + p ) − m/ for | λ | > R, for R = R ( m, k ˆ v k m ) > and then the proof follows. (cid:3) We restate [3, Lemma 2.6], which will be useful in section 4.
MATTEO SANTACESARIA
Lemma 2.3 ([3]) . Let a ∈ L s ( R ) ∩ L s ( R ) , < s < < s < ∞ and b ∈ L s ( R ) , < s < . Assume u is a function in L ˜ s ( R ) , with ˜ s defined asin (2.3) , which satisfies (2.20) ∂u ( λ ) ∂ ¯ λ = a ( λ )¯ u ( λ ) + b ( λ ) , λ ∈ C . Then there exists c > such that (2.21) k u k L ˜ s ≤ c k b k L s exp( c ( k a k L s + k a k L s )) . We will make also use of the well-known Hölder’s inequality, which werecall in a special case: for f ∈ L p ( C ) , g ∈ L q ( C ) such that ≤ p, q ≤ ∞ , ≤ r < ∞ , /p + 1 /q = 1 /r , we have k f g k L r ( C ) ≤ k f k L p ( C ) k g k L q ( C ) . From Φ to h ( λ ) Lemma 3.1.
Let the condition (1.8) holds. Then we have, for p ≥ , (cid:13)(cid:13)(cid:13)(cid:13) h ( λ )¯ λ (cid:13)(cid:13)(cid:13)(cid:13) L p ( | λ | >R ) ≤ c ( p, m ) k ˆ v k m R m +1 − /p , (3.1) k h k L p ( | λ | >R ) ≤ c ( p, m ) k ˆ v k m R m − /p , (3.2) where R is as in Lemma 2.2.Proof. It’s a corollary of Lemma 2.2. Indeed we have (cid:13)(cid:13)(cid:13)(cid:13) h ( λ )¯ λ (cid:13)(cid:13)(cid:13)(cid:13) pL p ( | λ | >R ) ≤ c k ˆ v k pm Z r>R r − mp − p dr = c ( p, m ) k ˆ v k pm R ( m +1) p − , (3.3)which gives (3.1). The proof of (3.2) is analogous. (cid:3) Lemma 3.2.
Let D ⊂ { x ∈ R : | x | ≤ l } , v , v be two potentials satisfy-ing (1.3) , (1.8) , (1.10) , let Φ , Φ the corresponding Dirichlet-to-Neumannoperator and h , h the corresponding generalised scattering amplitude. Let k v j k m, ≤ N , j = 1 , . Then we have (3.4) | h ( λ ) − h ( λ ) | ≤ c ( D, N ) e l | λ | k Φ − Φ k H / → H − / , λ ∈ C . Proof.
We have the following identity:(3.5) h ( λ ) − h ( λ ) = Z ∂D ψ ( z, λ )(Φ − Φ ) ψ ( z, λ ) | dz | , where ψ j ( z, λ ) are the Faddeev functions associated to the potential v j , j =1 , . This identity is a particular case of the one in [20, Theorem 1]: we referto that paper for a proof. EW GLOBAL STABILITY IN 2D 9
From this identity we have: | h ( λ ) − h ( λ ) | ≤ k ψ ( · , λ ) k H / ( ∂D ) k Φ − Φ k H / → H − / k ψ ( · , λ ) k H / ( ∂D ) . (3.6)Now take ˜ p > and use the trace theorem to get k ψ j ( · , λ ) k H / ( ∂D ) ≤ C k ψ j ( · , λ ) k W , ˜ p ( D ) ≤ Ce l | λ | k e − izλ ψ j ( · , λ ) k W , ˜ p ( D ) ≤ Ce l | λ | (cid:16) k e − izλ ψ j ( · , λ ) − k W , ˜ p ( D ) + k k W , ˜ p ( D ) (cid:17) , j = 1 , , which from (2.7) and (2.11) is bounded by C ( D, N ) e l | λ | . These estimatestogether with (3.6) give (3.4). (cid:3) The main results of this section are the following propositions:
Proposition 3.3.
Let v , v be two potentials satisfying (1.3) , (1.8) , (1.10) ,let Φ , Φ the corresponding Dirichlet-to-Neumann operator and h , h thecorresponding generalised scattering amplitude. Let < ε < , < p < − ε and k v j k m, ≤ N , j = 1 , . Then there exists a constant c = c ( D, N, m, p ) such that (3.7) (cid:13)(cid:13)(cid:13)(cid:13) h ( λ ) − h ( λ )¯ λ (cid:13)(cid:13)(cid:13)(cid:13) L p ( C ) ≤ c log(3 + k Φ − Φ k − H / → H − / ) − ( m +1 − /p ) . Proposition 3.4.
Let v , v , Φ , Φ , h , h be as in Proposition 3.3. Let p ≥ and k v j k m, ≤ N , j = 1 , . Then there exists a constant c = c ( D, N, m, p ) such that (3.8) k h − h k L p ( C ) ≤ c log(3 + k Φ − Φ k − H / → H − / ) − ( m − /p ) . Proof of Proposition 3.3.
Let choose a, b > , a close to and b big to bedetermined and let(3.9) δ = k Φ − Φ k H / → H − / . We split down the left term of (3.7) as follows: (cid:13)(cid:13)(cid:13)(cid:13) h ( λ ) − h ( λ )¯ λ (cid:13)(cid:13)(cid:13)(cid:13) L p ( C ) ≤ (cid:13)(cid:13)(cid:13)(cid:13) h ( λ ) − h ( λ )¯ λ (cid:13)(cid:13)(cid:13)(cid:13) L p ( | λ | b ) . From (2.12) we obtain (cid:13)(cid:13)(cid:13)(cid:13) h ( λ ) − h ( λ )¯ λ (cid:13)(cid:13)(cid:13)(cid:13) L p ( | λ | b ) ≤ c ( N ) b m +1 − /p . We now define(3.13) a = log(3 + δ − ) − m +1 − /pε − /p , b = β log(3 + δ − ) , for < β < / (2 l ) , in order to have (3.10) and (3.12) of the order log(3 + δ − ) − ( m +1 − /p ) . We also choose ¯ δ < such that for every δ ≤ ¯ δ , a issufficiently small in order to have (2.12) (which yields (3.10)), b ≥ R (with R as in Lemma 2.2) and also(3.14) δa − /p = δ log(3 + δ − ) (cid:16) m +1 − /pε − /p (cid:17) (1 − /p ) < log(3 + δ − ) − ( m +1 − /p ) . Thus we obtain (cid:13)(cid:13)(cid:13)(cid:13) h ( λ ) − h ( λ )¯ λ (cid:13)(cid:13)(cid:13)(cid:13) L p ( C ) ≤ c ( D, N, p )log(3 + δ − ) m +1 − /p (3.15) + c ( D, N ) δ (3 + δ − ) lβ , for δ ≤ ¯ δ , < β < / (2 l ) . As δ (3 + δ − ) lβ → for δ → more rapidly thanthe other term, we obtain that(3.16) (cid:13)(cid:13)(cid:13)(cid:13) h ( λ ) − h ( λ )¯ λ (cid:13)(cid:13)(cid:13)(cid:13) L p ( C ) ≤ c ( D, N, m, p, β )log(3 + δ − ) m +1 − /p , for δ ≤ ¯ δ , < β < / (2 l ) .Estimate (3.16) for general δ (with modified constant) follows from (3.16)for δ ≤ ¯ δ and the property (2.10) of the scattering amplitude. This completesthe proof of Proposition 3.3. (cid:3) Proof of Proposition 3.4.
We follow almost the same scheme as in the proofof Proposition 3.3. Let choose b > big to be determined and let(3.17) δ = k Φ − Φ k H / → H − / . We split down the left term of (3.8) as follows: k h − h k L p ( C ) ≤ k h − h k L p ( | λ | R , where R is defined in Lemma 2.2.Then we have, for δ ≤ ¯ δ , k h − h k L p ( C ) ≤ c ( D, N, m, p ) δ (1 + δ − ) lβ ( β log(3 + δ − )) /p + c ( N, m, p )(log(3 + δ − )) − ( m − /p ) . Since lβ < , we have that δ (1 + δ − ) lβ ( β log(3 + δ − )) /p → for δ → more rapidly than the other term. Thus(3.20) k h − h k L p ( C ) ≤ c ( D, N, m, p, β )(log(3 + δ − )) − ( m − /p ) , for δ ≤ ¯ δ , < β < / (2 l ) .Estimate (3.20) for general δ (with modified constant) follows from (3.20)for δ ≤ ¯ δ and the L p -boundedness of the scattering amplitude (this becauseit is continuous and decays at infinity like in Lemma 3.1). This completesthe proof of Proposition 3.4. (cid:3) Estimates on the Faddeev functions
Lemma 4.1.
Let v , v be two potentials satisfying (1.3) , (1.8) , (1.10) , with k v j k m, ≤ N , h , h the corresponding scattering amplitude and µ ( z, λ ) , µ ( z, λ ) the corresponding Faddeev functions. Let < s < , and ˜ s be as in (2.3) .Then sup z ∈ C k µ ( z, · ) − µ ( z, · ) k L ˜ s ( C ) ≤ c ( D, N, s ) (cid:13)(cid:13)(cid:13)(cid:13) h ( λ ) − h ( λ )¯ λ (cid:13)(cid:13)(cid:13)(cid:13) L s ( C ) , (4.1) sup z ∈ C (cid:13)(cid:13)(cid:13)(cid:13) ∂µ ( z, · ) ∂z − ∂µ ( z, · ) ∂z (cid:13)(cid:13)(cid:13)(cid:13) L ˜ s ( C ) ≤ c ( D, N, s ) " (cid:13)(cid:13)(cid:13)(cid:13) h ( λ ) − h ( λ )¯ λ (cid:13)(cid:13)(cid:13)(cid:13) L s ( C ) (4.2) + k h − h k L s ( C ) Proof.
We begin with the proof of (4.1). Let ν ( z, λ ) = µ ( z, λ ) − µ ( z, λ ) . (4.3) From the ¯ ∂ -equation (2.8) we deduce that ν satisfies the following non-homogeneous ¯ ∂ -equation: ∂∂ ¯ λ ν ( z, λ ) = e − λ ( z )4 π (cid:18) h ( λ )¯ λ ν ( z, λ ) + h ( λ ) − h ( λ )¯ λ µ ( z, λ ) (cid:19) , (4.4)for λ ∈ C , where e − λ ( z ) is defined in (2.9). Note that since, by Sobolevembedding, v ∈ L ∞ ( D ) ⊂ L s ( D ) , we have that ν ( z, · ) ∈ L ˜ s ( C ) for every ˜ s > (see (2.11)). In addition, from Proposition 2.1 (see (2.10)) we havethat h ( λ ) / ¯ λ ∈ L p ( C ) , for < p < ∞ . Then it is possible to use Lemma 2.3in order to obtain k ν ( z, · ) k L ˜ s ≤ c ( D, N, s ) (cid:13)(cid:13)(cid:13)(cid:13) µ ( z, λ ) h ( λ ) − h ( λ )¯ λ (cid:13)(cid:13)(cid:13)(cid:13) L s ( C ) ≤ c ( D, N, s ) sup z ∈ C k µ ( z, · ) k L ∞ (cid:13)(cid:13)(cid:13)(cid:13) h ( λ ) − h ( λ )¯ λ (cid:13)(cid:13)(cid:13)(cid:13) L s ( C ) ≤ c ( D, N, s ) (cid:13)(cid:13)(cid:13)(cid:13) h ( λ ) − h ( λ )¯ λ (cid:13)(cid:13)(cid:13)(cid:13) L s ( C ) , where we used again the property (2.11) of µ ( z, λ ) .Now we pass to (4.2). To simplify notations we write, for z, λ ∈ C , µ jz ( z, λ ) = ∂µ j ( z, λ ) ∂z , µ j ¯ z ( z, λ ) = ∂µ j ( z, λ ) ∂ ¯ z , j = 1 , . From the ¯ ∂ -equation (2.8) we have that µ jz and µ j ¯ z satisfy the followingsystem of non-homogeneous ¯ ∂ -equations, for j = 1 , : ∂∂ ¯ λ µ jz ( z, λ ) = e − λ ( z )4 π h j ( λ )¯ λ (cid:16) µ j ¯ z ( z, λ ) − iλµ j ( z, λ ) (cid:17) ,∂∂ ¯ λ µ j ¯ z ( z, λ ) = e − λ ( z )4 π h j ( λ )¯ λ (cid:16) µ jz ( z, λ ) − i ¯ λµ j ( z, λ ) (cid:17) . Define now µ j ± ( z, λ ) = µ jz ( z, λ ) ± µ j ¯ z ( z, λ ) , for j = 1 , . Then they satisfy thefollowing two non-homogeneous ¯ ∂ -equations: ∂∂ ¯ λ µ j ± ( z, λ ) = ± e − λ ( z )4 π h j ( λ )¯ λ (cid:16) µ j ± ( z, λ ) ∓ i ( λ ± ¯ λ ) µ j ( z, λ ) (cid:17) . Finally define τ ± ( z, λ ) = µ ± ( z, λ ) − µ ± ( z, λ ) . They satisfy the two non-homogeneous ¯ ∂ -equations below: ∂∂ ¯ λ τ ± ( z, λ ) = ± e − λ ( z )4 π (cid:20) h ( λ )¯ λ τ ± ( z, λ ) + h ( λ ) − h ( λ )¯ λ µ ± ( z, λ ) ∓ i λ ± ¯ λ ¯ λ (cid:16) ( h ( λ ) − h ( λ )) µ ( z, λ ) + h ( λ ) ν ( z, λ ) (cid:17) (cid:21) , where ν ( z, λ ) was defined in (4.3). EW GLOBAL STABILITY IN 2D 13
Now remark that by [19, Lemma 2.1] and regularity assumptions on thepotentials we have that µ jz ( z, · ) , µ j ¯ z ( z, · ) ∈ L ˜ s ( C ) ∩ L ∞ ( C ) for any ˜ s > , j = 1 , . This, in particular, yields τ ± ( z, · ) ∈ L ˜ s ( C ) . These arguments, alongwith the above remarks on the L p boundedness of h j ( λ ) / ¯ λ , make possible touse Lemma 2.3, which gives k τ ± ( z, · ) k L ˜ s ( C ) ≤ c ( D, N, s ) " (cid:13)(cid:13)(cid:13)(cid:13) h ( λ ) − h ( λ )¯ λ µ ± ( z, · ) (cid:13)(cid:13)(cid:13)(cid:13) L s ( C ) + k ( h ( · ) − h ( · )) µ ( z, · ) k L s ( C ) + k h ( · ) ν ( z, · ) k L s ( C ) ≤ c ( D, N, s ) " (cid:13)(cid:13)(cid:13)(cid:13) h ( λ ) − h ( λ )¯ λ (cid:13)(cid:13)(cid:13)(cid:13) L s ( C ) + k h − h k L s ( C ) + k h k L ( C ) k ν ( z, · ) k L ˜ s ( C ) ≤ c ( D, N, s ) " (cid:13)(cid:13)(cid:13)(cid:13) h ( λ ) − h ( λ )¯ λ (cid:13)(cid:13)(cid:13)(cid:13) L s ( C ) + k h − h k L s ( C ) , where we used Hölder’s inequality (since /s = 1 / / ˜ s ) and estimate(4.1). The proof of (4.2) now follows from this last inequality and the factthat µ z − µ z = ( τ + − τ − ) . (cid:3) Remark.
We also have proved that sup z ∈ C (cid:13)(cid:13)(cid:13)(cid:13) ∂µ ( z, · ) ∂ ¯ z − ∂µ ( z, · ) ∂ ¯ z (cid:13)(cid:13)(cid:13)(cid:13) L ˜ s ( C ) ≤ c ( D, N, s ) " (cid:13)(cid:13)(cid:13)(cid:13) h ( λ ) − h ( λ )¯ λ (cid:13)(cid:13)(cid:13)(cid:13) L s ( C ) + k h − h k L s ( C ) . We will need the following consequence of Lemma 4.1.
Lemma 4.2.
Let v , v be two potentials satisfying (1.3) , (1.8) , (1.10) , with k v j k m, ≤ N . Let h , h be the corresponding scattering amplitude and µ ( z, λ ) , µ ( z, λ ) the corresponding Faddeev functions. Let p, p ′ such that < p < < p ′ < ∞ , /p + 1 /p ′ = 1 . Then (4.5) k µ ( · , − µ ( · , k L ∞ ( D ) ≤ c ( D, N, p ) (cid:13)(cid:13)(cid:13)(cid:13) h ( λ ) − h ( λ )¯ λ (cid:13)(cid:13)(cid:13)(cid:13) L p ( C ) ∩ L p ′ ( C ) . Proof.
We recall again that if v ∈ W m, ( R ) , m > , with supp v ⊂ D then v ∈ L p ( D ) for p ∈ [1 , ∞ ] ; in particular, from Proposition 2.1, this yields h ( λ ) / ¯ λ ∈ L p ( C ) , for < p < ∞ .We write, as in the preceding proof, ν ( z, λ ) = µ ( z, λ ) − µ ( z, λ ) , (4.6) which satisfy the non-homogeneous ¯ ∂ -equations (4.4). From this equationwe obtain | ν ( z, | = 1 π (cid:12)(cid:12)(cid:12)(cid:12)Z C e − λ ( z )4 πλ h ( λ )¯ λ ν ( z, λ ) d Re λ d Im λ (4.7) + Z C e − λ ( z )4 πλ h ( λ ) − h ( λ )¯ λ µ ( z, λ ) d Re λ d Im λ (cid:12)(cid:12)(cid:12)(cid:12) ≤ π sup z ∈ C k ν ( z, · ) k L r (cid:13)(cid:13)(cid:13)(cid:13) h ( λ ) λ ¯ λ (cid:13)(cid:13)(cid:13)(cid:13) L r ′ + 14 π sup z ∈ C k µ ( z, · ) k L ∞ (cid:13)(cid:13)(cid:13)(cid:13) h ( λ ) − h ( λ ) λ ¯ λ (cid:13)(cid:13)(cid:13)(cid:13) L where /r + 1 /r ′ = 1 , < r ′ < < r < ∞ . The number s = 2 r/ ( r + 2) canbe chosen s < and as close to as wanted, by taking r big enough.Then (cid:13)(cid:13)(cid:13)(cid:13) h ( λ ) λ ¯ λ (cid:13)(cid:13)(cid:13)(cid:13) L r ′ ( | λ | Proof of Theorem 1.1. We begin with a remark, which take inspiration fromProblem 1 at non-zero energy (see, for instance, [18]).Let v ( z ) be a potential which satisfies the hypothesis of Theorem 1.1 and µ ( z, λ ) the corresponding Faddeev functions. Since µ ( z, λ ) satisfies (2.11),the ¯ ∂ -equation (2.8) and h ( λ ) decreases at infinity like in Lemma 2.2, it ispossible to write the following development:(5.1) µ ( z, λ ) = 1 + µ − ( z ) λ + O (cid:18) | λ | (cid:19) , λ → ∞ , for some function µ − ( z ) . If we insert (5.1) into equation (2.1), for ψ ( z, λ ) = e izλ µ ( z, λ ) , we obtain, letting λ → ∞ ,(5.2) v ( z ) = 4 i ∂µ − ( z ) ∂ ¯ z , z ∈ C . We can write this in a more explicit form, using the following integral equa-tion (a consequence of (2.8)): µ ( z, λ ) − π i Z C h ( λ ′ )( λ ′ − λ )¯ λ ′ e − λ ′ ( z ) µ ( z, λ ′ ) dλ ′ d ¯ λ ′ . By Lebesgue’s dominated convergence (using (2.12)) we obtain µ − ( z ) = − π i Z C h ( λ )¯ λ e − λ ( z ) µ ( z, λ ) dλ d ¯ λ, and the explicit formula(5.3) v ( z ) = 12 π Z C e − λ ( z ) ih ( λ ) µ ( z, λ ) − h ( λ )¯ λ (cid:18) ∂µ ( z, λ ) ∂z (cid:19)! dλ d ¯ λ. Formula (5.3) for v and v yields v ( z ) − v ( z ) = 12 π Z C e − λ ( z ) " i ( h ( λ ) − h ( λ )) µ ( z, λ )+ ih ( λ )( µ ( z, λ ) − µ ( z, λ )) − h ( λ ) − h ( λ )¯ λ (cid:18) ∂µ ( z, λ ) ∂z (cid:19) − h ( λ )¯ λ (cid:18) ∂µ ( z, λ ) ∂z − ∂µ ( z, λ ) ∂z (cid:19) dλ d ¯ λ. Then, using several times Hölder’s inequality, we find | v ( z ) − v ( z ) | ≤ π k µ ( z, · ) k L ∞ k h − h k L + k h k L ˜ p ′ k µ ( z, · ) − µ ( z, · ) k L ˜ p + (cid:13)(cid:13)(cid:13)(cid:13) h ( λ ) − h ( λ )¯ λ (cid:13)(cid:13)(cid:13)(cid:13) L p (cid:13)(cid:13)(cid:13)(cid:13) ∂µ ( z, · ) ∂z (cid:13)(cid:13)(cid:13)(cid:13) L p ′ + (cid:13)(cid:13)(cid:13)(cid:13) h ( λ )¯ λ (cid:13)(cid:13)(cid:13)(cid:13) L ˜ p ′ (cid:13)(cid:13)(cid:13)(cid:13) ∂µ ( z, · ) ∂z − ∂µ ( z, · ) ∂z (cid:13)(cid:13)(cid:13)(cid:13) L ˜ p ! , for < p < , ˜ p defined as in (2.3) and /p + 1 /p ′ = 1 / ˜ p + 1 / ˜ p ′ = 1 . From(2.11), (2.10), the continuity of h j and Lemma 2.2, [19, Lemma 2.1] (see theend of the proof of Lemma 4.1 for more details), Lemma 4.1, Propositions3.4 and 3.3 we finally obtain k v − v k L ∞ ( D ) ≤ c ( D, N, m, p ) (cid:18) log(3 + k Φ − Φ k − H / → H − / ) − ( m − + log(3 + k Φ − Φ k − H / → H − / ) − ( m +1 − /p ) + log(3 + k Φ − Φ k − H / → H − / ) − ( m − /p ) (cid:19) ≤ c ( D, N, m, p ) log(3 + k Φ − Φ k − H / → H − / ) − ( m − . This finishes the proof of Theorem 1.1. (cid:3) Proof of Theorem 1.2. We first extend σ on the whole plane by putting σ ( x ) = 1 for x ∈ R \ D (this extension is smooth by our hypothesis on σ ). Now since σ j | ∂D = 1 and ∂σ j ∂ν | ∂D = 0 for j = 1 , , from (1.5) we deducethat(5.4) Φ j = Λ j , j = 1 , . In addition, from (2.13) we get(5.5) lim λ → µ j ( z, λ ) = σ / j ( z ) , j = 1 , thus we obtain, using the fact that σ j is bounded from above and below, for j = 1 , , k σ − σ k L ∞ ( D ) ≤ c ( N ) k σ / − σ / k L ∞ ( D ) (5.6) = c ( N ) k µ ( · , − µ ( · , k L ∞ ( D ) . Now fix α < m and take p such that max (cid:18) , m − α + 1 (cid:19) < p < . EW GLOBAL STABILITY IN 2D 17 From Lemma 4.2 we have(5.7) k µ ( · , − µ ( · , k L ∞ ( D ) ≤ c ( D, N, p ) (cid:13)(cid:13)(cid:13)(cid:13) h ( λ ) − h ( λ )¯ λ (cid:13)(cid:13)(cid:13)(cid:13) L p ( C ) ∩ L p ′ ( C ) , where /p + 1 /p ′ = 1 . From Proposition 3.3 (cid:13)(cid:13)(cid:13)(cid:13) h ( λ ) − h ( λ )¯ λ (cid:13)(cid:13)(cid:13)(cid:13) L p ( C ) ∩ L p ′ ( C ) ≤ c ( D, N, p ) log(3 + k Φ − Φ k − H / → H − / ) − ( m +1 − /p ) ≤ c ( D, N, p ) log(3 + k Φ − Φ k − H / → H − / ) − α = c ( D, N, p ) log(3 + k Λ − Λ k − H / → H − / ) − α , from (5.4) and since α < m + 1 − p . Theorem 1.2 is thus proved. (cid:3) References [1] G. 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