Stability and uniqueness for a two-dimensional inverse boundary value problem for less regular potentials
aa r X i v : . [ m a t h . A P ] A p r Stability and uniqueness for a two-dimensionalinverse boundary value problem for lessregular potentials
E. Bl˚asten ∗ , O. Yu. Imanuvilov † , and M. Yamamoto ‡ September 27, 2018
Abstract
We consider inverse boundary value problems for the Schr¨odingerequations in two dimensions. Within less regular classes of poten-tials, we establish a conditional stability estimate of logarithmic order.Moreover we prove the uniqueness within L p -class of potentials with p > In this paper, we prove stability estimates and the uniqueness for an in-verse boundary value problem for the two-dimensional Schr¨odinger equationwithin a class of less regular unknown potentials. We refer to the first resultSylvester and Uhlmann [18] in the case where dimensions are higher thanor equal to three, and since then many remarkable works concerning theuniqueness have been published. Here we do not intend to create a completelist of publications and see e.g., a survey by Uhlmann [19]. In particular, thearguments in two dimensions are different from higher dimensions and werefer to the uniqueness result by Nachman [14], and a stability estimate byAlessandrini [2]. Also see Liu [11], and as survey on the uniqueness mainly intwo dimensions, see Imanuvilov and Yamamoto [8]. So far all these estimateshave had a logarithmic modulus of continuity, which is no surprise because ∗ Department of Mathematics, Tallinn University of Technology, Ehitajate tee 5, 19086Tallinn, Estonia, e-mail: [email protected] † Department of Mathematics, Colorado State University, 101 Weber Building, FortCollins, CO 80523-1874, U.S.A., e-mail: [email protected]
Partially supportedby NSF grant DMS 1312900 ‡ Department of Mathematical Sciences, The University of Tokyo, Komaba, Meguro,Tokyo 153, Japan, e-mail: [email protected] W p as pointedout in Bl˚asten’s licentiate thesis [5]. See also Novikov and Santacesaria [15],which proved stability assuming some smoothness and [16] which showed alsoa reconstruction formula. Santacesaria [17] continued working on stability,and showed that the smoother it is, the better exponent there will be on thelogarithm.There are not many results about stability and uniqueness for less regularpotentials and we refer to Bl˚asten [6], and Imanuvilov and Yamamoto [9].The former is the doctoral thesis of the first named author and proved condi-tional stability under some a priori boundedness of unknown potentials, andthe latter proved the uniqueness in determining L p -potentials with p > L p potentials, p >
2, and in addition give logarithmic type stability estimatesfor potentials in the class W s , s ∈ (0 , \ { } . After [6] and [9], the authorsrecognized that an improvement and simplification of the proofs are possible.That is, the main purpose of this paper is to improve the stability estimatesobtained in [6] and simplify the proof of [9] by using a unified method.The paper is composed of six sections. In Section 2, we formulate ourinverse problem and in Section 3 we state two main results Theorems 2.1 onthe conditional stability and Theorem 2.2 on the uniqueness and comparethem with the results in [6] and [9]. Sections 3-6 are devoted for completingthe proofs of Theorems 2.1 and 2.2. Let X ⊂ R be a bounded domain with boundary ∂X of C ∞ -class. Althoughit is possible to relax the regularity of the boundary for example to a Lipschitzdomain, we assume C ∞ -boundary for simplicity. Moreover let q ∈ L p ( X ), p >
2, be a potential function. Consider the Schr¨odinger operator with thepotential q in the domain XL q ( x, D ) u := ∆ u + qu. We define define the
Cauchy data C q by2 efinition 1.1. Let X ⊂ R be a bounded domain with smooth boundary ∂X and q ∈ L p ( X ) with p > . Then C q = { ( u, ∂ ν u ) ∈ W / ( ∂X ) × W − / ( ∂X ); L q ( x, D ) u = 0 , u ∈ W ( X ) } . If zero is not an eigenvalue of the operator L q ( x, D ) with the zero Dirichletboundary conditions, then the Cauchy data are equivalent to the Dirichlet-to-Neumann map Λ q defined byΛ q f = ∂u∂ν | ∂X , f ∈ W / ( ∂X ) , where u ∈ W ( X ) is a unique solution to L q ( x, D ) u = 0 in X and u | ∂X = f .The paper is concerned with a variant of the classical Calder´on problem: Suppose that for two potentials q and q the corresponding Cauchy data areequal. Does that imply the uniqueness of the potentials? The inverse problem asks whether the mapping q
7→ C q is invertible. Theuniqueness means that no two different potentials q have the same Cauchydata C q . The stability means that the mapping inverse to q
7→ C q is continu-ous in some topologies. For formulating the stability, we define the differenceof Cauchy data by d ( C q , C q ) := sup ( u ,u ) ∈X q ×X q (cid:12)(cid:12)(cid:12)(cid:12)Z X u ( q − q ) u dx (cid:12)(cid:12)(cid:12)(cid:12) , where X q = { u ∈ W ( X ); L q ( x, D ) u = 0 , k u k W ( X ) = 1 } . The difference d ( C q , C q ) is not a metric, but if C q = C q then d ( C q , C q ) = 0.Moreover if zero is not an eigenvalue of the operator L q j ( x, D ), j = 1 , d ( C q , C q ) ≤ C k Λ q − Λ q k L ( W / ( ∂X ); W − / ( ∂X )) by Lemma 3.2 proved below. Here the right-hand side denotes the operatornorm. This inequality means that for given C q and C q , without knowing q , q in X , it is possible to calculate an upper bound for d ( C q , C q ).Usually one can show only conditional stability , which means stabilityunder some assumptions on norms of unknown potentials q ’s. Other impor-tant topic is the reconstruction of a potential. That is, given a Cauchy data,reconstruct the potential using an explicit algorithm, and an even more valu-able goal is to reconstruct q in a stable way by given noisy data about C q .As for the reconstruction of less regular potentials, see Astala, Faraco and3ogers [4], which shows a reconstruction formula for potentials in W / , andproves that there exists a set of positive measure where the reconstructiondoes not converge pointwise for less regular potentials. Our proof suggeststhat the reconstruction converges in the L -norm and we here do not discussdetails. Notations.
Let i = √− x = ( x , x ) , x , x ∈ R , z = x + ix and z denote the complex conjugate of z ∈ C . We identify x ∈ R with z = x + ix ∈ C and ξ = ( ξ , ξ ) with ζ = ξ + iξ . We set ∂ z = ( ∂ x − i∂ x ), ∂ ¯ z = ( ∂ x + i∂ x ) . By L ( Y , Y ) we denote the space of linear continuousoperators from a Banach space Y into a Banach space Y . Let B (0 , δ ) bea ball in R of radius δ centered at 0 . We define the Fourier transform by( F u )( ξ ) = R R u ( x ) e − i ( x,ξ ) dx . Henceforth
C > X andconstants s, M , but independent of parameters τ , where s, M, τ are givenlater.We here state our two main results. Theorem 2.1.
Let X ⊂ R be a bounded domain with smooth boundary ∂X and s ∈ (0 , \ { } . We assume that q , q ∈ W s ( X ) satisfy an a prioriestimate k q j k W s ( X ) ≤ M with M < ∞ and q − q ∈ ˚ W s ( X ) . Then thereexists a constant C > such that k q − q k L ( X ) ≤ ( C (cid:16) d ( C q , C q ) (cid:17) − s/ , if d ( C q , C q ) < ,Cd ( C q , C q ) , if d ( C q , C q ) ≥ . Note that when s < no boundary behaviour is required from the twopotentials (e.g., Adams and Fournier [1], Lions and Magenes [10]).In our stability result, we estimate the norm k q − q k L ( X ) under the apriori boundedness of the norm in ˚ W s ( X ), while the work [6] uses differentnorms for q − q and a priori boundedness and for the norm. As for theexponent in the estimate, our result asserts − s/ − s/ k q − q k L ( X ) = O (cid:18) ln 1 d ( C q , C q ) (cid:19) − s/ ! d ( C q , C q ) −→
0. Thus the rate of the conditional stability is logarithmic.By Lemma 3.2 below, from Theorem 2.1, we can derive
Corollary.
Under the same assumptions of Theorem 2.1, we further assumethat zero is not an eigenvalue of L q j ( x, D ) with the zero Dirichlet boundarycondition. Let s ∈ (0 , , and let q , q ∈ W s ( X ) satisfy k q j k W s ( X ) ≤ M with M < ∞ and q − q ∈ ˚ W s ( X ) . Then there exists a constant C > such that k q − q k L ( X ) ≤ ( C (cid:16) k Λ q − Λ q k (cid:17) − s/ , if k Λ q − Λ q k < ,C k Λ q − Λ q k , if k Λ q − Λ q k ≥ . where k Λ q − Λ q k is the norm in L ( W / ( ∂X ); W − / ( ∂X )) . Our second main result is the uniqueness in the recovery of the potentialfor the Schr¨odinger operator :
Theorem 2.2.
Let X ⊂ R be a bounded smooth domain and q , q ∈ L p ( X ) with p > . If C q = C q , then q = q . The merits for the proof of our unified method are as follows.1. The proofs of both stability and uniqueness are simplified. Bl˚asten [6]used Sobolev spaces where the L p -norm has been replaced by a Lorentz-norm. We can avoid using the Lorentz-norm by showing a Carlemanestimate formulated using conventional L p -spaces.2. Comparing with Imanuvilov and Yamamoto [9], we use a simpler L -convergent stationary-phase argument which avoids approximating thepotentials by test functions and using Egorov’s theorem. We start this section with the following Lemma:
Lemma 3.1.
Let X ⊂ R be a bounded Lipschitz domain and q , q ∈ L p ( X ) , p > . If C q = C q , then Z X u ( q − q ) u dx = 0 for all ( u , u ) ∈ X q × X q . emma 3.2. Let X ⊂ R be a bounded smooth domain and q , q ∈ L p ( X ) , p > be potentials. We assume that is not an eigenvalue of the operator L q j ( x, D ) , j = 1 , , with the zero Dirichlet boundary condition. Then d ( C q , C q ) ≤ k Tr k L ( W ( X ); W / ( ∂X )) k Λ q − Λ q k L ( W / ( ∂X ); W − / ( ∂X )) . Proof.
Let u , U ∈ W ( X ) satisfy L q ( x, D ) U = L q ( x, D ) u = 0 in X and U = u on ∂X . Then∆( U − u ) + q ( U − u ) + ( q − q ) u = 0 in X and U − u = 0 on ∂X . Multiplying by u , integrating by parts and using∆ u + q u = 0 in X and U − u = 0 on ∂X , we have Z X u ( q − q ) u dx = Z ∂X ∂ ν ( u − U ) u dσ. Now note that (
U, ∂ ν U ) ∈ C q and ( u , ∂ ν u ) ∈ C q . This observation allowsus to switch to the Dirichlet-to-Neumann maps, and so (cid:12)(cid:12)(cid:12)(cid:12)Z ∂X ( ∂ ν u − ∂ ν U ) u dσ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z ∂X (Λ q u − Λ q U ) u dσ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z ∂X ((Λ q − Λ q ) u ) u dσ (cid:12)(cid:12)(cid:12)(cid:12) because u = U on ∂X. Now take the supremum over ( u , u ) ∈ X q × X q , to obtainsup ( u ,u ) ∈X q ×X q (cid:12)(cid:12)(cid:12)(cid:12)Z X u ( q − q ) u dx (cid:12)(cid:12)(cid:12)(cid:12) = sup ( u ,u ) ∈X q ×X q (cid:12)(cid:12)(cid:12)(cid:12)Z ∂X ((Λ q − Λ q ) u ) u dσ (cid:12)(cid:12)(cid:12)(cid:12) ≤ k Tr k L ( W ( X ); W / ( ∂X )) k Λ q − Λ q k L ( W / ( ∂X ); W − / ( ∂X )) . The proof of Lemma 3.2 is complete.Henceforth we identify z = x + ix ∈ C with x = ( x , x ) ∈ R .The following lemma plays the important role in the proof of Theorems2.1 and 2.2. Lemma 3.3.
Let τ > , ≤ s ≤ and Q ∈ W s ( R ) , z ∈ C . Then (cid:13)(cid:13)(cid:13)(cid:13) Q − Z R τπ e ± iτ ( ( z − z ) +( z − z ) ) Qdx (cid:13)(cid:13)(cid:13)(cid:13) L ( R ; dx ) ≤ τ − s/ k Q k W s ( R ) . (1) If s = 0 , then the left-hand side tends to as τ → ∞ . roof. First for δ >
0, we have θ δ ( ξ ) := F ( e ± iτ ( z + z ) − δ | z | )( ξ )= π √ δ + 4 τ exp (cid:18) − δ | ξ | τ + 4 δ (cid:19) exp (cid:18) ∓ iτ ( ξ − ξ ) τ τ + 2 δ (cid:19) . The calculations are direct and we refer to pp.210-211 in Evans [7] for exam-ple. Let S ( R ) be the space rapidly decreasing functions and S ′ ( R ) be thedual, that is, the space of tempered distributions. Since θ δ −→ π τ exp (cid:18) ∓ i ( ξ − ξ )8 τ (cid:19) = π τ exp ∓ i ( ζ + ζ )16 τ ! and e ± iτ ( z + z ) − δ | z | −→ e ± iτ ( z + z ) as δ ↓ S ′ ( R ) and F is continuous from S ′ ( R ) to itself, we see F ( e ± iτ ( z + z ) )( ξ ) = π τ exp ∓ i ( ζ + ζ )16 τ ! in S ′ ( R ). This equality holds for almost all ξ ∈ R , because the right-handside is in L ∞ ( R ).Next let Q ∈ C ∞ ( R ) be arbitrarily chosen. Then F (cid:18) τπ e ± iτ ( z + z ) ∗ Q (cid:19) = exp ∓ i ( ζ + ζ )16 τ ! F ( Q )( ξ ) . Hence by the Plancherel theorem, we have (cid:13)(cid:13)(cid:13) Q − τπ e ± iτ ( z + z ) ∗ Q (cid:13)(cid:13)(cid:13) L ( R ) = π (cid:13)(cid:13)(cid:13) F Q − F (cid:16) τπ e ± iτ ( z + z ) ∗ Q (cid:17)(cid:13)(cid:13)(cid:13) L ( R ) = π (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) − e ∓ i ξ ξ τ (cid:19) F Q (cid:13)(cid:13)(cid:13)(cid:13) L ( R ) . On the other hand, we can prove | − e ∓ i ( ζ + ζ ) | ≤ s/ | ξ | s for 0 ≤ s ≤ ζ ∈ C . In fact, if | ξ | ≥
1, then | − e ∓ i ( ζ + ζ ) | ≤ ≤ s/ and so the inequality is seen. Let | ξ | ≤
1. Direct calculations yield | − e ∓ i ( ζ + ζ ) | = 4 sin ( ξ − ξ ). Therefore | − e ∓ i ( ζ + ζ ) | ≤ | ξ − ξ | ≤ | ξ + ξ | ≤ × s | ξ | s , ≤ s ≤ | ξ | ≤
1. Thus we have seen | − e ∓ i ( ξ + ξ ) | ≤ s/ | ξ | s for 0 ≤ s ≤ ξ ∈ C .Hence (cid:13)(cid:13)(cid:13)(cid:13) Q − τπ e ± iτ ( z + z ) ∗ Q (cid:13)(cid:13)(cid:13)(cid:13) L ( R ) ≤ π s/ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ξ p | τ | ! s F Q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( R ) ≤ π s/ − s | τ | − s/ k (1 + | ξ | ) s/ F Q k L ( R ) . (2)for each Q ∈ C ∞ ( R ). Since C ∞ ( R ) is dense in W s ( R ), passing to thelimits, we complete the proof of Lemma 3.3 for s > . If s = 0 and Q ∈ L ( R )for any positive ǫ we take a function Q ǫ ∈ C ∞ ( R ) such that k Q − Q ǫ k L ( R ) ≤ ǫ. Then (2) implies that for any positive τ (cid:13)(cid:13)(cid:13)(cid:13) Q − Q ǫ − τπ e ± iτ ( z + z ) ∗ ( Q − Q ǫ ) (cid:13)(cid:13)(cid:13)(cid:13) L ( R ) ≤ π k Q − Q ǫ k L ( R ) ≤ ǫ. Then applying to the function Q ǫ estimate (1), we obtain the statement ofour lemma for s = 0 . Let us introduce the operators:¯ ∂ − g = − π Z X g ( ξ , ξ ) ζ − z dξ dξ , ∂ − g = − π Z X g ( ξ , ξ ) ζ − z dξ dξ , where X ⊂ R is a bounded domain with the smooth boundary.We have Proposition 4.1. A ) Let ≤ p ≤ and < γ < p − p . Then ¯ ∂ − , ∂ − ∈L ( L p ( X ) , L γ ( X )) . B )Let < p < ∞ . Then ¯ ∂ − , ∂ − ∈ L ( L p ( X ) , W p ( X )) . A) is proved on p.47 in [20] and B) can be verified by using Theorem 1.32(p.56) in [20]. (cid:4)
Henceforth for arbitrarily fixed z ∈ C , we setΦ( z ) = Φ( z ; z ) := ( z − z ) and introduce the operator: e R τ g = 12 e − iτ (Φ+Φ) ∂ − ( ge iτ (Φ+Φ) ) .
8e set U = 1 , U = e R τ ( 12 ( ¯ ∂ − q − ¯ ∂ − q ( x ))) , (3) U j = e R τ ( 12 ¯ ∂ − ( qU j − )) ∀ j ≥ . (4)We construct a solution to the Schr¨odinger equation in the form u = ∞ X j =0 e iτ Φ ( − j U j . (5)Henceforth C ( ǫ ) denotes generic constants which are dependent on notonly s, M, X but also ǫ .We will prove that the infinite series is convergent in L r ( X ) with some r >
2. For it, we show the following propositions.
Proposition 4.2.
Let u ∈ W p ( X ) for any p > . Then for any ǫ ∈ (0 , there exists a constant C ( ǫ ) independent of x ∈ X and τ such that τ − ǫ k e R τ u k L ( X ) + τ /p k e R τ u k L ∞ ( X ) ≤ C ( ǫ ) k u k W p ( X ) ∀ τ > . (6) Proof.
Let ρ ∈ C ∞ ( B (0 , ρ | B (0 , ) = 1 . We set ρ τ = ρ ( √ τ ( x − x )) . Since e R τ u = e R τ ( ρ τ u ) + e R τ ((1 − ρ τ ) u ) for any positive ǫ , there exists p ( ǫ ) > k e iτ (Φ+¯Φ) ρ τ u k L p ǫ ) ( X ) ≤ C ( ǫ ) k u k W p ( X ) /τ − ǫ . Moreoversince k e iτ (Φ+¯Φ) u k L ∞ ( X ) ≤ C k u k W p ( X ) we have k e iτ (Φ+¯Φ) ρ τ u k L ∞ ( X ) ≤ C ( ǫ ) k u k W p ( X ) /τ − ǫ . Hence applying Proposition 4.1 and the Sobolev embedding theorem, we have τ − ǫ k e R τ ( ρ τ u ) k L ( X ) + τ /p k e R τ ( ρ τ u ) k L ∞ ( X ) ≤ C ( ǫ ) k u k W p ( X ) , ∀ ǫ ∈ (0 , . (7)Observe that Z X (1 − ρ τ ) ue iτ (Φ+Φ) z − ζ dξ = Z X (1 − ρ τ ) u∂e iτ (Φ+Φ) τ ( z − ζ ) i∂ Φ dξ = Z ∂X ( ν − iν )(1 − ρ τ ) ue iτ (Φ+Φ) iτ ( z − ζ ) ∂ Φ dσ − Z X τ ( z − ζ ) ∂ (cid:18) (1 − ρ τ ) ui∂ Φ (cid:19) e iτ (Φ+Φ) dξ + (1 − ρ τ ) ue iτ (Φ+Φ) iτ ∂ Φ . (8)9bviously, by the Sobolev embedding theorem, for any positive ǫ , thereexists a constant C ( ǫ ) such that τ − ǫ (cid:13)(cid:13)(cid:13)(cid:13) (1 − ρ τ ) uτ ∂ Φ (cid:13)(cid:13)(cid:13)(cid:13) L ( X ) + τ / (cid:13)(cid:13)(cid:13)(cid:13) (1 − ρ τ ) uτ ∂ Φ (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( X ) ≤ C ( ǫ ) k u k W p ( X ) . (9)For the second term on the right-hand side of (8), we have (cid:12)(cid:12)(cid:12)(cid:12)Z X τ ( z − ζ ) ∂ (cid:18) (1 − ρ τ ) u∂ Φ (cid:19) e iτ (Φ+Φ) dξ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ ( z − ζ ) (cid:18) ∂ρ ( √ τ ξ ) u∂ Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dξ + Z X (cid:12)(cid:12)(cid:12)(cid:12) τ ( z − ζ ) (cid:18) (1 − ρ τ ) ∂u∂ Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dξ + Z X (cid:12)(cid:12)(cid:12)(cid:12) τ ( z − ζ ) (cid:18) (1 − ρ τ ) u ( ∂ Φ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dξ. The functions (1 − ρ τ ) ∂u∂ Φ are uniformly bounded in τ in L p ( X ) for some p ∈ (1 , . Moreover, since k (1 − ρ τ ) /∂ Φ k L ∞ ( X ) ≤ C √ τ , the functions √ τ (1 − ρ τ ) ∂u∂ Φ are uniformly bounded in τ in functions (1 − ρ τ ) ∂u √ τ∂ Φ are uniformly bounded in τ in L p ( X ) . Applying Proposition 4.1, we have τ (cid:13)(cid:13)(cid:13)(cid:13) ∂ − (cid:18) (1 − ρ τ ) ∂ z uτ ∂ Φ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ( X ) + τ /p (cid:13)(cid:13)(cid:13)(cid:13) ∂ − (cid:18) (1 − ρ τ ) ∂ z uτ ∂ Φ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( X ) ≤ C k u k W p ( X ) . (10)On the other hand, for any p > (cid:13)(cid:13)(cid:13)(cid:13) ∂ρ ( √ τ · ) u∂ Φ (cid:13)(cid:13)(cid:13)(cid:13) L p ( X ) ≤ C k u k C ( X ) (cid:13)(cid:13)(cid:13)(cid:13) ∂ Φ (cid:13)(cid:13)(cid:13)(cid:13) L p ( B (0 , √ τ )) ≤ Cτ (2 − p ) / p k u k W p ( X ) . Thanks to this inequality, applying Proposition 4.1 again, we have: τ − ǫ (cid:13)(cid:13)(cid:13)(cid:13) τ ∂ − (cid:18) ∂ρ ( √ τ · ) u∂ Φ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ( X ) + τ /p (cid:13)(cid:13)(cid:13) τ ∂ − (cid:16) ∂ρ ( √ τ · ) u∂ Φ (cid:17)(cid:13)(cid:13)(cid:13) L ∞ ( X ) ≤ C ( ǫ ) k u k W p ( X ) . (11)For any p >
1, we have (cid:13)(cid:13)(cid:13)(cid:13) (1 − ρ τ ) u ( ∂ Φ) (cid:13)(cid:13)(cid:13)(cid:13) L p ( X ) ≤ C k u k C ( X ) (cid:13)(cid:13)(cid:13)(cid:13) ∂ Φ) (cid:13)(cid:13)(cid:13)(cid:13) L p ( X \ B (0 , √ τ )) ≤ C ( p ) k u k W p ( X ) τ (2 p − / p . Therefore τ − ǫ (cid:13)(cid:13)(cid:13)(cid:13) ∂ − (cid:18) (1 − ρ τ ) uτ ( ∂ Φ) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ( X ) + τ /p (cid:13)(cid:13)(cid:13) ∂ − (cid:16) (1 − ρ τ ) uτ ( ∂ Φ) (cid:17)(cid:13)(cid:13)(cid:13) L ∞ ( X ) ≤ C ( ǫ ) k u k W p ( X ) . (12)10rom the classical representation of the Cauchy integral (see e.g. [13]p.27) we obtain (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z ∂X ( ν − iν )(1 − ρ τ ) ue iτ (Φ+Φ) iτ ( z − ζ ) ∂ Φ dσ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( X ) ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ( ν − iν )(1 − ρ τ ) ue iτ (Φ+Φ) iτ ∂ Φ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( ∂X ) ≤ C (cid:13)(cid:13)(cid:13)(cid:13) (1 − ρ τ ) ∂ Φ (cid:13)(cid:13)(cid:13)(cid:13) L ( ∂X ) k u k W p ( X ) /τ ≤ C k u k W p ( X ) ln τ /τ. (13)By the trace theorem and the Sobolev embedding theorem, for any p > α = α ( p ) such that the trace operator is continuousfrom W p ( X ) into C α ( ∂X ) . Using Theorem 1.11 (see p. 22 of [20]), for any δ ∈ (0 , α ( p )), there exists a constant C ( δ ) > (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z ∂X ( ν − iν )(1 − ρ τ ) ue iτ (Φ+Φ) iτ ( z − ζ ) ∂ Φ dσ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( X ) ≤ C ( δ ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ( ν − iν )(1 − ρ τ ) ue iτ (Φ+Φ) iτ ∂ Φ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C δ ( ∂X ) ≤ C ( δ ) k (1 − ρ τ ) ∂ Φ e iτ (Φ+Φ) k C δ ( ∂X ) k u k W p ( X ) /τ. Denote µ τ ( x ) = (1 − ρ τ ) ∂ Φ e iτ (Φ+Φ) . Then by the definitions of the functions Φand ρ τ (noting that we identify z with x ), we estimate k µ τ ( · ) k C ( ∂X ) ≤ C √ τ and k∇ µ τ ( · ) k C ( ∂X ) ≤ Cτ ∀ τ > . Since in view of the mean value theorem, we can estimate | µ τ ( x ) − µ τ ( x ′ ) | = | µ τ ( x ) − µ τ ( x ′ ) | − δ | µ τ ( x ) − µ τ ( x ′ ) | δ ≤ Cτ − δ τ δ | x − x ′ | δ (14)and we obtain (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z ∂X ( ν − iν )(1 − ρ τ ) ue iτ (Φ+Φ) iτ ( z − ζ ) ∂ Φ dσ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( X ) ≤ C ( δ ) k u k W p ( X ) /τ (1 − δ ) / . (15)From (7)-(15) we have (6). (cid:4) L r ( X ) for all sufficiently large τ. Let ˜ p ∈ (2 , p ) . By (6) and Proposition 4.1and the H¨older inequality, there exists a positive constant δ (˜ p ) such that k e R τ u k L p ˜ pp − ˜ p ( X ) ≤ C k u k W p ( X ) /τ δ . (16)Using (16) we have k U j k L p ˜ pp − ˜ p ( X ) ≤ Cτ δ k
12 ¯ ∂ − ( qU j − ) k W p ( X ) ≤ C τ δ k ¯ ∂ − k L ( L ˜ p ( X ); W p ( X )) k qU j − k L ˜ p ( X ) ≤ C τ δ k ¯ ∂ − k L ( L ˜ p ( X ); W p ( X )) k q k L p ( X ) k U j − k L ˜ ppp − ˜ p ( X ) ≤ C k ¯ ∂ − k L ( L ˜ p ( X ); W p ( X )) k q k L p ( X ) τ δ ! j − k U k L p ˜ pp − ˜ p ( X ) . (17)Therefore there exists τ such that for all τ > τ k U j k L p ˜ pp − ˜ p ( X ) ≤ j k U k L p ˜ pp − ˜ p ( X ) ∀ j ≥ . Hence the convergence of the series is proved.Since L q ( x, D )( U j e iτ Φ ) = 4 ¯ ∂∂ ( e iτ Φ e R τ ( 12 ¯ ∂ − ( qU j − ))) + qU j e iτ Φ = 2 ¯ ∂ ( e iτ Φ
12 ¯ ∂ − ( qU j − )) + q U j e iτ Φ = qU j − e iτ Φ + qU j e iτ Φ , the infinite series (5) represents the solution to the Schr¨odinger equation. ByProposition 4.2, we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j =2 ( − j U j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( X ) = O (cid:18) τ (cid:19) as τ → + ∞ . (18)Besides the estimate (18) we need the estimate of the infinite series P ∞ j =2 ( − j U j in the space L ∞ ( X ) . By Proposition 4.2, we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j =2 ( − j U j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( X ) = O (cid:18) τ p (cid:19) as τ → + ∞ . (19)12 roposition 4.3. Let q ∈ L p ( X ) and < p < ∞ . Then there exists apositive constant b C ( k q k L p ( X ) ) independent of τ and x such that if τ > b C ( k q k L p ( X ) ) and x ∈ X , then there exists u ∈ W ( X ) such that L q ( x, D ) u =0 in X and u ( x, x ) = e iτ Φ (1 − e − iτ (¯Φ+Φ) ∂ − ( e iτ (¯Φ+Φ) ( ¯ ∂ − q − ¯ ∂ − q ( x )))+ r ( x, x )) , (20) and there exists a positive constant C , independent of τ and x ∈ X , suchthat τ sup x ∈ X k r ( · , x ) k L ( X ) + τ + p sup x ∈ X k r ( · , x ) k L ( X ) ≤ C k q k L p ( X ) , (21) k u k W ( X ) ≤ C e R τ , (22) whenever | x | < R where R > is large enough that X ⊂ B (0 , R ) . Proof.
Above we proved that the infinite series (5) for all sufficientlylarge τ is the solution to the equation L q ( x, D ) u = 0 . We set r ( x, x ) = P ∞ j =2 ( − j U j . Thanks to (3) we have (20). The estimate of the first term in(21) follows from (18). By (18) and (19), we havesup x ∈ X k r ( · , x ) k L ( X ) ≤ sup x ∈ X k r ( · , x ) k L ( X ) sup x ∈ X k r ( · , x ) k L ∞ ( X ) ≤ C k q k L p ( X ) τ − p τ − /p ≤ C k q k L p ( X ) τ + p . (23)Finally estimate (22) follows from (20), (21) and the classical estimate forelliptic equations. (cid:4) We set τ = max { b C ( k q k L p ( X ) ) b C ( k q k L p ( X ) ) } , where b C ( k q k k L p ( X ) ) are deter-mined in Proposition 4.3 and let τ ≥ τ such that it is larger than τ fromProposition 4.3. For point x ∈ X and τ ≥ τ let u ∈ W ( X ) be the solutionto L q ( x, D ) u = 0 given by Proposition 4.3. In particular we have u ( x, x ) = e iτ Φ (1 − e − iτ (¯Φ+Φ) ∂ − ( e iτ (¯Φ+Φ) ( ¯ ∂ − q − ¯ ∂ − q ( x ))) + r ( x, x )) , (24)sup x ∈ X k r ( · , x ) k L ( X ) τ + sup x ∈ X k r ( · , x ) k L ( X ) τ + p ≤ C k q k L p ( X ) , (25)13up x ∈ X k u ( · , x ) k W ( X ) ≤ Ce R τ , (26)and there exists a solution u ∈ W ( X ) for L q ( x, D ) u = 0 with u ( x, x ) = e iτ Φ (1 − e − iτ (¯Φ+Φ) ¯ ∂ − ( e iτ (¯Φ+Φ) ( ∂ − q − ∂ − q ( x ))) + r ( x, x )) , (27)sup x ∈ X k r ( · , x ) k L ( X ) τ + sup x ∈ X k r ( · , x ) k L ( X ) τ + p ≤ C k q k L p ( X ) , (28)sup x ∈ X k u ( · , x ) k W ( X ) ≤ Ce R τ , (29)where constant C is independent of τ and x . Substituting (24) and (27) into R X u ( q − q ) u dx and using the Fubini theorem on the Cauchy-operators,we obtain( q − q )( x ) = (cid:18) ( q − q )( x ) − Z X τπ e iτ (Φ+Φ) ( q − q )( x ) dx (cid:19) + 2 τπ Z X u ( q − q ) u dx − τπ Z X ¯ ∂ − ( q − q )( ∂ − q − ∂ − q ( x )) e iτ (¯Φ+Φ) dx − τπ Z X ∂ − ( q − q )( ¯ ∂ − q − ¯ ∂ − q ( x )) e iτ (¯Φ+Φ) dx − τπ Z X e iτ (Φ+Φ) ( q − q )( x )( p p + r + r )( x, x ) dx, (30)where p = r − e − iτ (Φ+Φ) ∂ − ( e iτ (Φ+Φ) ( ∂ − q − ∂ − q ( x ))) , (31) p = r − e − iτ (Φ+Φ) ¯ ∂ − ( e iτ (Φ+Φ) ( ∂ − q − ∂ − q ( x ))) . (32)We recall that q − q ∈ ˚ W s ( X ) by the assumptions of the theorem. For s ∈ (0 , \ (cid:8) (cid:9) and q ∈ ˚ W s ( X ), let E q be the extension in R by the zeroextension outside X . Then E q ∈ W s ( R ).We can now deal with the first term. Take the L ( X )-norm with respectto x to obtain (cid:13)(cid:13)(cid:13)(cid:13) q − q − Z X τπ e iτ (Φ+Φ) ( q − q )( x ) dx (cid:13)(cid:13)(cid:13)(cid:13) L ( X : ,dx ) = (cid:13)(cid:13)(cid:13)(cid:13) E ( q − q ) − Z R τπ e iτ (Φ+Φ) E ( q − q )( x ) dx (cid:13)(cid:13)(cid:13)(cid:13) L ( R ; dx ) . (cid:13)(cid:13)(cid:13)(cid:13) q − q − Z X τπ e iτ (Φ+Φ) ( q − q )( x ) dx (cid:13)(cid:13)(cid:13)(cid:13) L ( X ; dx ) ≤ τ − s/ k E ( q − q ) k W s ( R ) ≤ Cτ − s/ k q − q k W s ( X ) ≤ CM τ − s/ . (33)The second term on the right-hand side of (30) is estimated by the dif-ference of the boundary data and the definition of d ( C q , C q ): (cid:13)(cid:13)(cid:13)(cid:13) τπ Z X u ( q − q ) u dx (cid:13)(cid:13)(cid:13)(cid:13) L ( X ; dx ) ≤ C sup x ∈ X (cid:12)(cid:12)(cid:12)(cid:12) τπ Z X u ( q − q ) u dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cτ d ( C q , C q ) sup x ∈ X ( k u k W ( X ) k u k W ( X ) ) ≤ C M e τ (8 R +1) d ( C q , C q ) . (34)Here in order to obtain the last estimate, we used (26) and (29). ApplyingLemma 3.3 again, we obtain that there exists ˜ s > (cid:13)(cid:13)(cid:13)(cid:13) τπ Z X ¯ ∂ − ( q − q )( ∂ − q − ∂ − q ( x )) e iτ (¯Φ+Φ) dx (cid:13)(cid:13)(cid:13)(cid:13) L ( X ; dx ) (35) ≤ (cid:13)(cid:13)(cid:13)(cid:13) ¯ ∂ − ( q − q ) ∂ − q − τπ Z X ¯ ∂ − ( q − q ) ∂ − q e iτ (¯Φ+Φ) dx (cid:13)(cid:13)(cid:13)(cid:13) L ( X ; dx ) + (cid:13)(cid:13)(cid:13)(cid:13) ¯ ∂ − ( q − q ) ∂ − q − ∂ − q τπ Z X ¯ ∂ − ( q − q ) e iτ (¯Φ+Φ) dx (cid:13)(cid:13)(cid:13)(cid:13) L ( X ; dx ) ≤ (cid:13)(cid:13)(cid:13)(cid:13) E ¯ ∂ − ( q − q ) E ∂ − q − τπ Z R E ¯ ∂ − ( q − q ) E ∂ − q e iτ (¯Φ+Φ) dx (cid:13)(cid:13)(cid:13)(cid:13) L ( R ; dx ) + (cid:13)(cid:13)(cid:13)(cid:13) E ¯ ∂ − ( q − q ) E ∂ − q − E ∂ − q τπ Z R E ¯ ∂ − ( q − q ) e iτ (¯Φ+Φ) dx (cid:13)(cid:13)(cid:13)(cid:13) L ( R ; dx ) ≤ Cτ ˜ s k E ¯ ∂ − ( q − q ) E ∂ − q k W ( R ) + Cτ ˜ s k E ¯ ∂ − ( q − q ) k W ( R ) k ∂ − q k L ∞ ( X ) ≤ C ′ τ ˜ s k q − q k L ( X ) . In a similar way we obtain (cid:13)(cid:13)(cid:13)(cid:13) τπ Z X ∂ − ( q − q )( ¯ ∂ − q − ¯ ∂ − q ( x )) e iτ (¯Φ+Φ) dx (cid:13)(cid:13)(cid:13)(cid:13) L ( X ; dx ) ≤ C ′ τ ˜ s k q − q k L ( X ) . (36)15stimating the L -norm of the last term on the righ-hand side of (30), wehave I = (cid:13)(cid:13)(cid:13)(cid:13) τπ Z X e iτ (Φ+Φ) ( q − q )( x )( p p + r + r )( x, x ) dx (cid:13)(cid:13)(cid:13)(cid:13) L ( X ; dx ) ≤ C sup x ∈ X τπ Z X | ( q − q )( x ) || ( p p + r + r )( x, x ) | dx. Thanks to (25) and (28), we obtain
I ≤ Cτ k q − q k L ( X ) sup x ∈ X k ( p p + r + r )( · , x ) k L ( X ) ≤ Cτ k q − q k L ( X ) sup x ∈ X ( k p p k L ( X ) + k ( r + r )( · , x ) k L ( X ) ) ≤ C k q − q k L ( X ) sup x ∈ X ( τ k p p k L ( X ) + 1 √ τ ) . By (25), (28) and Proposition 4.3sup x ∈ X k p p k L ( X ) ≤ sup x ∈ X ( k r k L ( X ) k r k L ( X ) + 14 k ∂ − ( e iτ (Φ+Φ) ( ∂ − q − ∂ − q ( x ))) k L ∞ ( X ) k r k L ( X ) + 14 k ∂ − ( e iτ (Φ+Φ) ( ∂ − q − ∂ − q ( x ))) k L ∞ ( X ) k r k L ( X ) + 116 k ∂ − ( e iτ (Φ+Φ) ( ∂ − q − ∂ − q ( x ))) k L ( X ) k ∂ − ( e iτ (Φ+Φ) ( ∂ − q − ∂ − q ( x ))) k L ∞ ( X ) ) ≤ C (cid:16) τ + 1 τ p ( k r k L ( X ) + k r k L ( X ) )+ 1 τ p k ∂ − ( e iτ (Φ+Φ) ( ∂ − q − ∂ − q ( x ))) k L ( X ) (cid:17) . Applying (25), (28) and Proposition 4.2 with ǫ = p , we obtain:sup x ∈ X k p p k L ( X ) ≤ C ( 1 τ + 1 τ p ( 1 τ + 1 τ − p )) . (37)Hence there exists τ independent of z such that I ≤ k q − q k L ( X ) ∀ τ ≥ τ . (38)16ombining estimates (33)-(38) and setting R = 8 R + 1, we obtain k q − q k L ( X ) ≤ C ( e τR d ( C q , C q ) + τ − s/ ) , ∀ τ ≥ τ . (39)Replacing τ and C by τ + τ and Ce R τ respectively, we have (39) for all τ >
0. For obtaining the conditional stability, we should make the right-handside of (39) as small as possible by choosing τ >
0. For this we make thefollowing choice of τ depending on the value of d ( C q , C q ) . Case 1: d ( C q , C q ) < τ = αR (cid:18) d ( C q , C q ) (cid:19) > α ∈ (0 , e τR d ( C q , C q ) = e α d ( C q , C q ) − α and τ − s/ = (cid:18) R α (cid:19) s/ (cid:18) d ( C q , C q ) (cid:19) − s/ . Since for 0 < α <
1, there exists a constant
C > η − α ≤ C (cid:16) η (cid:17) − s/ for 0 ≤ η <
1, with this choice of τ , estimate (39) yields k q − q k L ( X ) ≤ C (cid:18) d ( C q , C q ) (cid:19) − s/ . Case 2: d ( C q , C q ) ≥ k q k W s ( X ) ≤ M and k q k W s ( X ) ≤ M , we have k q − q k L ( X ) ≤ M ≤ M d ( C q , C q ).Therefore combining the two cases, we complete the proof of Theorem2.1. (cid:4) For any point x ∈ X let u , u ∈ W ( X ) be the solutions to the Schr¨odingerequation given by (24) and (27) respectively.Since the Dirichlet-to-Neumann maps are the same, we have R X ( q − q ) u u dx = 0. Then plugging formulas (24) and (27) into it and adding( q − q )( x ) to both sides, we have 17 q − q )( x ) = (cid:18) ( q − q )( x ) − Z X τπ e iτ (Φ+Φ) ( q − q )( x ) dx (cid:19) − τπ Z X ¯ ∂ − ( q − q )( ∂ − q − ∂ − q ( x )) e iτ (¯Φ+Φ) dx − τπ Z X ∂ − ( q − q )( ¯ ∂ − q − ¯ ∂ − q ( x )) e iτ (¯Φ+Φ) dx − τπ Z X e iτ (Φ+Φ) ( q − q )( x )( p p + r + r )( x, x ) dx, (40)where the functions p j are determined by (31) and (32).Since the estimates (35), (38) hold true for all sufficiently large τ , weobtain from (40): k q − q k L ( X ) ≤ C (cid:13)(cid:13)(cid:13)(cid:13) q − q − Z X τπ e iτ (Φ+Φ) ( q − q )( x ) dx (cid:13)(cid:13)(cid:13)(cid:13) L ( X ; dx ) = C (cid:13)(cid:13)(cid:13)(cid:13) E ( q − q ) − Z R τπ e iτ (Φ+Φ) E ( q − q )( x ) dx (cid:13)(cid:13)(cid:13)(cid:13) L ( R ; dx ) . In view of Lemma 3.3 we obtain (cid:13)(cid:13)(cid:13)(cid:13) q − q − Z X τπ e iτ (Φ+Φ) ( q − q )( x ) dx (cid:13)(cid:13)(cid:13)(cid:13) L ( X ; dx ) → τ → + ∞ . The proof of the theorem is complete. (cid:4)
Acknowledgement . The authors thank the anonymous referees forvaluable comments.
References [1] R.A. Adams and John J.F. Fournier,
Sobolev Spaces , Elsevier/AcademicPress, Amsterdam, 2003.[2] G. Alessandrini,
Stable determination of conductivity by boundary mea-surements , Appl. Anal., (1988), 153-172.[3] A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case , J. Inverse Ill-Posed Probl., (2008), 19–33.[4] K. Astala, D. Faraco and K.M. Rogers, Rough potential recovery in theplane , ArXiv e-prints, 2013, http://arxiv.org/abs/1304.1317.185] E. Bl˚asten,
The inverse problem of the Schr¨odinger equation in the plane:A dissection of Bukhgeim’s result , University of Helsinki, Licentiate the-sis, 2010, http://arxiv.org/abs/1103.6200.[6] E. Bl˚asten,
On the Gel’fand-Calder´on inverse problem in two dimen-sions , University of Helsinki, Doctoral thesis, 2013.[7] L.C. Evans,
Partial Differential Equations , Amer. Math. Soc., Provi-dence, Rhode Island, 1998.[8] O.Y. Imanuvilov and M. Yamamoto,
Uniqueness for inverse boundaryvalue problems by Dirichlet-to-Neumann map on subboundaries , MilanJ. Math., (2013), 187-258.[9] O.Y. Imanuvilov and M. Yamamoto, Inverse boundary value problemfor linear Schr¨odinger equation in two dimensions , ArXiv e-prints, 2012,http://adsabs.harvard.edu/abs/2012arXiv1208.3775I.[10] J.L. Lions and E. Magenes,
Non-homogeneous Boundary Value Problemsand Applications , vol.1, Springer-Verlag, Berlin, 1972.[11] L. Liu,
Stability estimates for the two dimensional inverse conductivityproblem , University of Rochester, Doctral thesis, 1997.[12] N. Mandache,
Exponential instability in an inverse problem for theSchr¨odinger equation , Inverse Problems, (2001), 1435–1444.[13] C. Miranda, Partial differential equations of elliptic type,
Second Re-vised Edition, Springer-Verlag, 1970.[14] A.I. Nachman,
Global uniqueness for a two-dimensional inverse bound-ary value problem , Ann. of Math., (1996), 71–96.[15] R.G. Novikov and M. Santacesaria,
A global stability estimate for theGel’fand-Calder´on inverse problem in two dimensions , J. Inverse Ill-Posed Probl., (2010), 765–785.[16] R.G. Novikov and M. Santacesaria, Global uniqueness and reconstructionfor the multi-channel Gel’fand-Calder´on inverse problem in two dimen-sions , Bull. Sci. Math., (2011) 421–434.[17] M. Santacesaria,
New global stability estimates for the Calder´on problemin two dimensions , J. Inst. Math. Jussieu, (2013), 553–569.1918] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverseboundary value problem , Ann. of Math., (1987), 153–169.[19] G. Uhlmann,