On uniqueness in the inverse conductivity problem with local data


 Annabella Mason
 1 years ago
 Views:
Transcription
1 On uniqueness in the inverse conductivity problem with local data Victor Isakov June 21, Introduction The inverse condictivity problem with many boundary measurements consists of recovery of conductivity coefficient a (principal part) of an elliptic equation in a domain R n, n = 2, 3 from the Neumann data given for all Dirichlet data (DirichlettoNeumann map). Calderon [5] proposed the idea of using complex exponential solutions to demonstrate uniqueness in the linearized inverse condictivity problem. Complex exponential solutions of elliptic equations have been introduced by Faddeev [7] for needs of inverse scattering theory. Sylvester and Uhlmann in their fundamental paper [19] attracted ideas from geometrical optics, constructed almost complex exponential solutions for the Schrödinger operator, and proved global uniqueness of a ( and of potential c in the Schrödinger equation) in the threedimensional case. In the twodimensional case the inverse conductivity problem is less overdetermined, and the Sylvester and Uhlmann method is not applicable, but one enjoys advantages of the methods of inverse scattering and of theory of complex variables. Using these methods Nachman [17] demonstrated uniqueness of a C 2 ( ) and Astala and Päivärinta [1] showed uniqueness of a L () which is a final result in the inverse conductivity problem in R 2 with many measurements from the whole boundary. There is a known hypothesis (see for example, [11], Problem 5.3, [14], [20]) that the DirichlettoNeumann map given at any (nonvoid open) part Γ of the boundary also uniquely determines conductivity coefficient or potential in the Schrödinger equation. This local boundary measurements model important 1
2 applications, for example to geophysics or to semiconductors when collecting data from the whole boundary is either not possible or extremely expensive. Despite extended long term efforts this hypothesis remains not proven, altough there is some progress. Kohn and Vogelius [15] showed uniqueness of the boundary reconstruction (of all existing partial derivatives of a) and hence uniqueness of piecewise analytic a. When coefficients are known in a neighborhood of the boundary, then Runge type approximation argument reduces the partial DirichlettoNeumann map to complete map [9], [11], Exercise 5.7.4, [15], and hence the hypothesis follows. In smooth case Bukhgeim and Uhlmann [4] made use of Carleman estimates (with linear phase function) to show that the Neumann data on a sufficiently large part Γ of the boundary given for all Dirichlet data on the whole boundary uniquely determine potential c in the threedimensional Schrödinger equation. The most advanced and recent result is due to Kenig, Sjöstrand and Uhlmann [14]. They modified the scheme of [19], [4] by using quadratic phase function and demonstrated uniqueness of c from Neumann data on Γ for all Dirichlet data on a complementary part Γ 1. While Γ in [14] can be arbitrarely small one can not assume zero Dirichlet data on \ Γ (although they can be zero on \ Γ 1 ) and they need Γ and Γ 1 to have nonvoid intersection. Only result concering zero boundary data on part Γ 0 of the boundary is due to Hähner [8] who by explicit calculations proved completeness of products of harmonic functions which are zero on a spherical Γ 0. An inverse scattering in halfspace was considered by Karamyan [13]. In this paper we give a complete proof of this hypothesis when the DirichlettoNeumann map is given on arbitrary part Γ of while on the remaining part Γ 0 one has homogeneous Dirichlet or Neumann data. Our restrictive assumption is that Γ 0 is a part of a plane or of a sphere. In some applications this assumption is natural, but available uniqueness results [6], [16] require that coefficients of the differential equation are known near. An exception is the paper [13] where one is given scattering data in halfspace. This assumption enables to reflect almost complex exponential solutions across Γ 0 and to avoid use of a special fundamental solution (Green s function) for the Schrödinger equation and of corresponding exponential weighted estimates with a large parameter in. Currently, such fundamental solution and estimates are available only when homogeneous boundary data are given at a part of the boundary (but not at the whole boundary) [4], [14]. It is not likely that such fundamenatl solutions and 2
3 estimates can be found when homogeneous Dirichlet (or Neumann) are prescribed at the whole boundary. We add new ingredients to the Sylvester Uhlmann method. A crucial observation is that contributions of products of almost complex exponential solutions and of their reflections converge to zero when large parameter τ goes to. 2 Main results Let be a domain in R 3 with Lipschitz boundary. We consider the Schrödinger equation u + cu = 0 in (2.1) with the Dirichlet boundary data or the Neumann data u = g 0 on (2.2) ν u = g 1 on. (2.3) Let B 0 be some ball. We will assume that the (complex valued) potential c L (), c = k 2 on \ B 0. Let Γ 0 be an open bounded part of and Γ = \ Γ 0. We define the local DirichlettoNeumann map Λ c (D, Γ) as Λ c (D, Γ)g 0 = ν u on Γ, g 0 H 1 2 ( ), g0 = 0 on Γ 0 and the local NeumanntoDirichlet map as Λ c (N, Γ)g 1 = u on Γ, g 1 H 1 2 ( ), g1 = 0 on Γ 0 (2.4) provided the Dirichlet or Neumann problems are uniquely solvable. In Theorems 2.1 and 2.2 we consider two cases and a) is a bounded subdomain of {x : x 3 < 0}, Γ = {x 3 < 0}; b) 0 < k, = {x : x 3 < 0} and Γ \ B. (2.5) It is well known that in case (2.5), a), the boundary value problems (2.1), (2.2) or (2.3) have unique solutions u H 1 () for any boundary data g 0 H 1 2 ( ), g 1 H 1 2 ( ) provided there is uniqueness of a solution. 3
4 Uniqueness is guaranteed by maximum principles or energy integrals when Ic = 0, 0 c on or when Ic 0 on a nonempty open subset of. In case (2.5), b), we will assume that Ic 0 on and we will augment the equation and the boundary condition by the Sommerfeld radiation condition lim r(σ u iku)(x) = 0, σ = r 1 x, as r = x. (2.6) By using integral equations or the LaxPhillips method one can demonstrate uniqueness and existence of a solution u H 1 ( B) for any ball B to the scattering boundary value problem (2.1), (2.2) or (2.3), and (2.6) with compactly supported g 0 H 1 2 ( ), g 1 H 1 2 ( ) [11], [16]. Theorem 2.1 If or Λ c1 (D, Γ) = Λ c2 (D, Γ) (2.7) Λ c1 (N, Γ) = Λ c2 (N, Γ), (2.8) then c 1 = c 2 This result has immediate corollary for the conductivity equation div(a u) k 2 u = 0 in. (2.9) We will assume that a C 2 ( ), a = 1 on \ B 0, a > 0 on, and 0 < k in case (2.5), b). As for the Schrödinger equation, in case (2.5), a), the elliptic boundary value problems (2.9), (2.2) or (2.3) are uniquely solvable in H 1 () for all k except discrete set of eigenvalues. When k = 0, the Dirichlet problem is uniquely solvable. In case (2.5), b) the boundary value scattering scattering problems (2.9), (2.2) or (2.3), (2.6) are uniquely solvable in the same functional spaces as for the Schrödinger equation. We will assume the unique solvability condition and we define local DirichlettoNeumann and NeumanntoDirichlet maps for the conductivity equation similarly to the Schrödinger equation and we will denote them by Λ(a; D, Γ), Λ(a; N, Γ). Theorem 2.2 If or Λ(a 1 ; D, Γ) = Λ(a 2 ; D, Γ) (2.10) Λ(a 1 ; N, Γ) = Λ(a 2 ; N, Γ) (2.11) 4
5 with the additional assumption then a 1 = a 2 on. 3 a 1 = 3 a 2 = 0 on Γ 0, (2.12) These results imply similar results for bounded domains when Γ 0 is a part of a sphere. In Theorems 2.3, 2.4 we use the following notation and assumptions. Let be a subdomain of B 0. Let Γ 0 = B 0 and Γ = \ Γ 0. We will assume that Γ 0 B 0. Theorem 2.3 The equality (2.7) implies that c 1 = c 2 Theorem 2.4 The equality (2.10) implies that a 1 = a 2 Finally we give available results the plane case and assume that is a bounded simply connected domain in R 2 with Lipschitz boundary. According to established theory of elliptic boundary value problems if a L (), k = 0, then the Dirichlet problem (2.9), (2.2) has a unique solution u H 1 () for any g 0 H 1 2 ( ), and the Neumann problem (2.9), (2.3) has a unique solution in the same space with normalization condition u = 0 provided g 1 = 0. We remind that the (conormal) derivative a ν u H 1 2 ( ) is defined as (a ν u)ϕ = a u ϕ, ϕ H 1 2 ( ), (2.13) where the integral on the left side is understood as dual pairing between H 1 2 ( ) and H 1 2 ( ). After these reminders we can define the partial DirichlettoNeumann and NeumanntoDirichlet maps as above. Theorem 2.5 Let a 1, a 2 L (). Let Γ be any nonvoid open arc of. Then equalities (2.10) and (2.11) imply that a 1 = a 2 in. Theorem 2.5 was proven by Astala, Lassas, and Päivärinta [2], Theorem 2.3. Theorems can be immediately generalized to semilinear Schrödinder equations and quasilinear conductivity equations as in [12], [18]. 5
6 3 Proofs for halfspace In this section we consider the case of {x : x 3 < 0} when unobservable part Γ 0 of the boundary is contained in the plane {x 3 = 0}. We give proofs for Neumann boundary condition, because of its applied importance, and we will indicate how to modify them for Dirichlet condition. We start with a standard orthogonality relation. Lemma 3.1 Under condition (2.8) (c 1 c 2 )v 1 v 2 = 0 (3.14) for all functions v 1, v 2 H 1 ( B), for any ball B, satisfying v 1 + c 1 v 1 = 0 in, ν v 1 = 0 on Γ 0 (3.15) and v 2 + c 2 v 2 = 0 in, ν v 2 = 0 on Γ 0. (3.16) Proof: First we consider case a) of bounded domain. Let v 1 be any solution to (3.15). Let u 2 be the solution to the Schrödinger equation with c = c 2 and with the Neumann data ν u 2 = ν v 1 on. Subtracting the equations u 2 + c 2 u 2 = 0 and v 1 + c 1 v 1 = 0 and letting v = u 2 v 1 we yield v + c 2 v = (c 1 c 2 )v 1 on. (3.17) We have v = 0 on Γ by condition (2.8) and (2.4) and ν v = 0 (by definition of v) on. Multiplying equation (3.17) by solution v 2 to (3.16) and integrating by parts we yield (c 1 c 2 )v 1 v 2 = ( v + c 2 v)v 2 = v( v 2 + c 2 v 2 ) = 0 where we used the boundary conditions ν v = 0 = ν v 2 on Γ 0 and the equality v = ν v = 0 on Γ. Summing up we have the orthogonality relation (3.14). 6
7 The case of = {x 3 < 0} needs in addition a Runge type approximation argument Let u 1 be any solution to (3.15). Let u 2 be the solution to the Schrödinger equation with c = c 2 and with the Neumann data ν u 2 = ν u 1 on. Subtracting the equations u 2 +c 2 u 2 = 0 and u 1 +c 1 u 1 = 0 and letting v = u 2 u 1 we yield v + c 2 v = (c 1 c 2 )u 1 on. (3.18) Since v = 0 by condition (2.8) and ν v = 0 (by definition of v) on Γ we have v = 0 on \ B 0 due to uniqueness in the Cauchy problem for the Laplace equation. Multiplying equation (3.18) by v 2 and integrating by parts we yield (c 1 c 2 )u 1 v 2 = B 0 ( v + c 2 v)v 2 = B 0 v( v 2 + c 2 v 2 ) = 0 B 0 where we used the boundary conditions ν v = 0 = ν v 2 on Γ 0 and the equality v = 0 on \ B 0. Summing up we have the orthogonality relation (3.14) for v 1 = u 1. To complete the proof we will L 2 ( B 0 )approximate arbitrary v 1 by u 1. Let us assume the opposite: the subspace {u 1 } is not dense in {v 1 }. Then by HahnBanach theorem there is f L 2 (), f = 0 outside B 0, such that for all u 1, but Let u 1(f) solve the Neumann problem From (3.19) and (3.21) we have fu 1 = 0 (3.19) fv 1 0 for some v 1. (3.20) u 1 + c 1 u 1 = f in, ν u 1 = 0 on. (3.21) 0 = ( u 1 + c 1 u 1)u 1 = Γ u 1 ν u 1 where we used the Green s formula and boundary conditions for u 1, u 1, Since ν u 1 can be arbitrary smooth function on Γ we conclude that u 1 is zero on Γ. Now u 1 solves the elliptic equation u 1 = 0 on \ B 0 and has zero 7
8 Cauchy data on Γ, so by uniqueness in the Cauchy problem u 1 = 0 on \B 0. Applying again the Green s formula in B 0 we yield which contradicts (3.20). The proof is complete. Lemma 3.2 Under condition (2.7) fv 1 = ( u 1 + c 1 u 1)v 1 = 0 B 0 (c 1 c 2 )v 1 v 2 = 0 for all functions v 1, v 2 H 1 ( B), for any ball B, satisfying v 1 + c 1 v 1 = 0 in, v 1 = 0 on Γ 0 and v 2 + c 2 v 2 = 0 in, v 2 = 0 on Γ 0 Proof is similar to the proof of Lemma 3.1. Proof of Theorem 2.1 First we remind known results about existence of special almost complex exponential solutions to the Schrödinger equation in R 3. Let ξ = (ξ 1, ξ 2, ξ 3 ), ξ = (ξ 1, ξ 2, ξ 3 ). We introduce e(1) = (ξ ξ 2 2) 1 2 (ξ1, ξ 2, 0), e(3) = (0, 0, 1), and the unit vector e(2) to get orthonormal basis e(1), e(2), e(3) in R 3. We denote the coordinates of x in this basis by (x 1e, x 2e, x 3e ) e. Observe that ξ = (ξ 1e, 0, ξ 3 ) e, ξ 1e = (ξ1 2 + ξ2) and that in general x y = x 1 y 1 + x 2 y 2 + x 3 y 3 = x 1e y 1e + x 2e y 2e + x 3e y 3e. 8
9 We define ζ(1) = ( ξ 1e 2 τξ 3, i ξ ( τ 2 ) 1 2, ξ τξ 1e) e, ζ (1) = ( ξ 1e 2 τξ 3, i ξ ( τ 2 ) 1 2, ξ 3 2 τξ 1e) e, ζ(2) = ( ξ 1e 2 + τξ 3, i ξ ( τ 2 ) 1 2, ξ 3 2 τξ 1e) e, ζ (2) = ( ξ 1e 2 + τξ 3, i ξ ( τ 2 ) 1 ξ 3 2, 2 + τξ 1e) e, (3.22) where τ is a positive real number. By direct calculations we can see that ζ(1) ζ(1) = ζ (1) ζ (1) = ζ(2) ζ(2) = ζ (2) ζ (2) = 0. (3.23) Let us extend c 1, c 2 onto R 3 as even functions of x 3. Since (3.23) holds, it is known [11], section 5.3, [19], that there are almost exponential solutions to the equations with e iζ(1) x (1 + w 1 ), e iζ(2) x (1 + w 2 ) u 1 + c 1 u 1 = 0, u 2 + c 2 u 2 = 0 in R 3 (3.24) w 1 2 (B 0 ) + w 2 2 (B 0 ) 0 as τ, (3.25) where 2 (B) is the standard norm in L 2 (B). We define f (x 1, x 2, x 3 ) = f(x 1, x 2, x 3 ) and we let u 1 (x) = e iζ(1) x (1 + w 1 ) + e iζ (1) x (1 + w 1), u 2 (x) = e iζ(2) x (1 + w 2 ) + e iζ (2) x (1 + w 2). (3.26) It is obvious that u 1, u 2 H 2 ( B) for any B, solve the partial differential equations (3.24) and that ν u 1 = ν u 2 = 0 on Γ 0. (3.27) Let c = c 1 c 2. (3.28) 9
10 By (3.24), (3.27), (3.28), and Lemma = cu 1 u 2 = c(x)(e i(ζ(1)+ζ(2)) x (1 + w 1 (x))(1 + w 2 (x))+ e i(ζ (1)+ζ(2)) x (1 + w 1(x))(1 + w 2 (x))+ e i(ζ(1)+ζ (2)) x (1 + w 1 (x))(1 + w 2(x))+ e i(ζ (1)+ζ (2)) x (1 + w 1(x))(1 + w 2(x)))dx due to (3.26). Using (3.22) we conclude that c(x)(e iξ x (1 + w 1 (x))(1 + w 2 (x))+ e i(ξ 1ex 1e 2τξ 1e x 3 ) (1 + w 1(x))(1 + w 2 (x))+ e i(ξ 1ex 1e +2τξ 1e x 3 ) (1 + w 1 (x))(1 + w 2(x))+ e iξ x (1 + w 1(x))(1 + w 2(x)))dx = 0. (3.29) Now we let τ. Observe that moduli of all exponents are bounded by 1. So due to (3.25) limits of all terms containing factors w j, wj are zero. By the RiemannLebesgue Lemma limits of c(x)e i(ξ 1ex 1e 2τξ 1e x 3 ), c(x)e i(ξ 1ex 1 +2τξ 1e x 3 ) dx as τ are also zero provided ξ 1e 0. Therefore from (3.29) we derive that c(x)(e iξ x + e iξ x )dx = 0 (3.30) for any ξ, ξ 1e 0. Since c j and hence c (given by (3.28) )are compactly supported, the right side in (3.30) is analytic with respect to ξ, so we have (3.30) for all ξ R 3. Since c is an even function of x 3, c(x)(e iξ x + e iξ x )dx = c(x)e iξ x dx. R 3 Hence from (3.30) R 3 c(x)e iξ x dx = 0 10
11 for any ξ R 3. By uniqueness of the inverse Fourier transformation c = 0, and hence c = 0 and c 1 = c 2. This completes the proof under condition (2.8). Now we will show how to adjust it to the case of Dirichlet boundary conditions. The argument until (3.25) is the same. Then we let u 1 (x) = e iζ(1) x (1 + w 1 (x)) e iζ (1) x (1 + w 1(x)), u 2 (x) = e iζ(2) x (1 + w 2 (x)) e iζ (2) x (1 + w 2(x)). (3.31) It is obvious that u 1, u 2 H 2 ( B 0 ), solve the partial differential equations (3.24) and that u 1 = u 2 = 0 on Γ 0. (3.32) By (3.24), (3.32) and Lemma = cu 1 u 2 = c(x)(e i(ζ(1)+ζ(2)) x (1 + w 1 (x))(1 + w 2 (x)) e i(ζ (1)+ζ(2)) x (1 + w 1(x))(1 + w 2 (x)) e i(ζ(1)+ζ (2)) x (1 + w 1 (x))(1 + w 2(x))+ e i(ζ (1)+ζ (2)) x (1 + w 1(x))(1 + w 2(x)))dx due to (3.31). Using (3.22) we conclude that c(x)(e iξ x (1 + w 1 (x))(1 + w 2 (x)) e i(ξ 1ex 1e 2τξ 1e x 3 ) (1 + w 1(x))(1 + w 2 (x)) e i(ξ 1ex 1e +2τξ 1e x 3 ) (1 + w 1 (x))(1 + w 2(x))+ e iξ x (1 + w 1(x))(1 + w 2(x)))dx = 0. As above we let τ and repeating the argument after (3.29) we conclude that 0 = c(x)(e iξ x + e iξ x )dx = c(x)e iξ x dx. R 3 11
12 for any ξ R 3. By uniqueness of the inverse Fourier transformation c = 0, and hence c 1 = c 2. The proof is complete. Proof of Theorem 2.2 The well known substitution u = a 1 2 v (3.33) transforms the conductivity equation (2.9) into the Schrödinger equation (2.1) with c = a a 2. (3.34) From (3.33) it follows that homogeneous boundary Dirichlet condition for u on Γ 0 implies the same condition for v and since 3 a = 0 on Γ 0 due to (2.12) the same is true for the Neumann condition. As known ([11]), section 5.2, ([15]) the local DirichlettoNeumann (NeumanntoDirichlet) map on Γ uniquely determines a, 3 a on Γ. Hence again (3.33) implies that partial DirichlettoNeumann map Λ(a; D, Γ) for the conductivity equation uniquely determines partial DirichlettoNeumann map Λ c (D, Γ) for the Schrödinger equations. The same holds for the NeumanntoDirichlet maps. By Theorem 2.1 the potential c given by (3.34) is unique. (3.28) can be viewed as an linear elliptic equation a ca 1 2 = 0 in with respect to a 1 2. As mentioned Λ(a; D, Γ) or Λ(a; D, Γ) uniquely determine c on and a, 3 a on Γ. So due to uniqueness in the Cauchy problem for elliptic equations ([11], section 3.3) a is uniquely determined on as well. The proof is complete. We observe that Theorem 2.2 holds for complex valid a with Ra > 0 which are of importance in applications [11], p.7. The above proof is valid with the substitution (3.33) where a 1 is the principal branch which is well 2 defined on the halfplane Ra > 0. 4 Proofs for subsets of balls: use of the Kelvin transform We remind the definition of the Kelvin transform of a function in R 3. Let X(x) = x 2 x, x(x) = X 2 X (4.35) 12
13 and It is known [10] that U(X) = X 1 u(x(x)) (4.36) X U(X) = X 5 x u(x(x)) (4.37) Proof of Theorem 2.3 We can assume that B is the ball of radius 1 2 centered at x 0 = (0, 0, 1 2 ) and that the origin in R 3 does not belong to. Let us apply the Kelvin transform to the equation (2.1). Using (4.37) and (4.36) we yield X U + CU = 0, where C(X) = X 6 c(x(x)) The inversion (4.35) transforms the sphere { x x 0 = 1 } into the plane 2 {X 3 = 1}. Hence the domain in Xvariables is a subdomain of the halfspace {X : 1 < X 3 }, and parts Γ 0, Γ of its boundary are correspondingly parts of the plane {X 3 = 1} and of the halfspace {1 < X 3 }. Due to (4.36) homogeneous Dirichlet data on Γ 0 are transformed into homogeneous Dirichlet data in new variables. Obviously, the DirichlettoNeumann map in xvariables uniquely determines the DirichlettoNeumann map on Γ in new variables. Applying Theorem 2.1 to the inverse problem in X variables we conclude that C(X) = X 6 c(x(x)) is uniquely determined. Hence c is uniquely determined on. The proof is complete. 5 Conclusion The main remaining open question is of course how obtain uniqueness from local DirichlettoNeumann map when Γ 0 is an arbitrary surface. When this map is given at all k uniqueness follows from the results on inverse hyperbolic problems obtained by the methods of boundary control [3]. In particular, it is important to relax topological assumptions on and to consider case of several connected components of, especially when Γ 0 is an inner connected component of the boundary. For spherical Γ 0 one can most likely to use methods of this paper. In case of halfspace by more careful study of bahavior at inifinity one can relax the assumtpion that c is compactly supported. We formulated results assuming that Dirichlet or Neumann problems are uniquely 13
14 solvable only to follow traditions. Arguments in the threedimensional case will not change if we drop these assumptions and instead require equalities of Cauchy pairs (u, a ν u) on Γ (as in [20], section 3), under homogeneous Dirichlet or Neumann conditions on Γ 0. It is very interesting and probably more difficult to recover simulteneously a, Γ 0 and coefficient b of the boundary condition a ν u + bu = 0 on Γ 0. Again such results are available for hyperbolic equations. It is still not clear if it is possible to construct semilocalized (in weighted spaces) almost complex exponential solutions to the Schrödinger equation and to use them in uniqueness proofs and constructive methods of solution of the inverse problem with local data or with unknown multiple inclusions. Localization would most likely remove all geometrical assumptions and to obtain most general results. Due to substantial overdeterminancy of this inverse problem it is feasible that a completely different approach will work and resolve remaining questions. Aknowledgement: This research was in part supported by the NSF grant DMS References [1] Astala, K., Päivärinta, L. Calderon s inverse conductivity problem in the plane. Ann. Math.., 163 (2006), [2] Astala, K., Lassa, M., Päivärinta, L. Calderon s inverse conductivity problem for anisotropic conductivity in the plane. Comm. Part. Diff. Equat., 30 (2005), [3] Belishev, M. Boundary Control in Reconstruction of Manifolds and Metrics (the BCmethod). Inverse Problems, 13 (1997), R1R45. [4] Bukhgeim, A.L., Uhlmann, G. Recovering a Potential from Partial Cauchy Data. Comm. Part. Diff. Equat., 27 (2002),
15 [5] Calderon, A.P. On an inverse boundary value problem. In Seminar on Numerical Analysis and Its Applications to Continuum Physics, Rio de Janeiro, (1980), [6] Eskin, G., Ralston, J. Inverse coefficient problems in perturbed half spaces. Inverse Problems, 15 (1999), [7] Faddeev, L.D. Growing solutions of the Schrödinger equation. Dokl. Akad. Nauk SSSR, 165 (1965), [8] Hähner, P. A Uniqueness Theorem for an Inverse Scattering Problem in an Exterior Domain. SIAM J. Math. Anal., 29 (1998), [9] Isakov, V. On uniqueness of recovery of a discontinuous conductivity coefficient. Comm. Pure Appl. Math, 41 (1988), [10] Isakov, V., Inverse Source Problems. AMS, Providence, RI, [11] Isakov, V., Inverse Problems for PDE. SpringerVerlag, New York, [12] Isakov, V., Sylvester, J. Global uniqueness for a semilinear elliptic inverse problem. Comm. Pure Appl. Math., 47 (1994), [13] Karamyan, G. Inverse scattering in a half space with passive boundary. Comm. Part. Diff. Equat., 28 (2003), [14] Kenig, C., Sjöstrand, J., Uhlmann, G. The Calderon problem with partial data. Ann. Math. (to appear). [15] Kohn, R., Vogelius, M. Determining conductivity by boundary measurements. II. Interior results. Comm. Pure Appl. Math., 38 (1985), [16] Lassas, M., Cheney, M., Uhlmann, G. Uniqueness for a wave propagation inverse problem in a halfspace. Inverse Problems, 14 (1998), [17] Nachman, A. A global uniqueness for a two dimensional inverse boundary value problem. Ann. Math., 142 (1995), [18] Sun, Z., On a Quasilinear Inverse Boundary Value Problem. Math. Z., 221 (1996),
16 [19] Sylvester, J., Uhlmann, G. Global uniqueness theorem for an inverse boundary value problem. Ann. Math., 125 (1987), [20] Uhlmann, G. Developments in inverse problems since Calderon s fundamental paper. Harmonic Analysis and PDE, Univ. of Chicago Press. Victor Isakov Department of Mathematics and Statistics Wichita State University Wichita, KS , U.S.A. 16
Complex geometrical optics solutions for Lipschitz conductivities
Rev. Mat. Iberoamericana 19 (2003), 57 72 Complex geometrical optics solutions for Lipschitz conductivities Lassi Päivärinta, Alexander Panchenko and Gunther Uhlmann Abstract We prove the existence of
More informationA CALDERÓN PROBLEM WITH FREQUENCYDIFFERENTIAL DATA IN DISPERSIVE MEDIA. has a unique solution. The DirichlettoNeumann map
A CALDERÓN PROBLEM WITH FREQUENCYDIFFERENTIAL DATA IN DISPERSIVE MEDIA SUNGWHAN KIM AND ALEXANDRU TAMASAN ABSTRACT. We consider the problem of identifying a complex valued coefficient γ(x, ω) in the conductivity
More informationInverse Gravimetry Problem
Inverse Gravimetry Problem Victor Isakov September 21, 2010 Department of M athematics and Statistics W ichita State U niversity W ichita, KS 67260 0033, U.S.A. e mail : victor.isakov@wichita.edu 1 Formulation.
More informationCOMPLEX SPHERICAL WAVES AND INVERSE PROBLEMS IN UNBOUNDED DOMAINS
COMPLEX SPHERICAL WAVES AND INVERSE PROBLEMS IN UNBOUNDED DOMAINS MIKKO SALO AND JENNNAN WANG Abstract. This work is motivated by the inverse conductivity problem of identifying an embedded object in
More informationMonotonicity arguments in electrical impedance tomography
Monotonicity arguments in electrical impedance tomography Bastian Gebauer gebauer@math.unimainz.de Institut für Mathematik, Joh. GutenbergUniversität Mainz, Germany NAMKolloquium, GeorgAugustUniversität
More informationSurvey of Inverse Problems For Hyperbolic PDEs
Survey of Inverse Problems For Hyperbolic PDEs Rakesh Department of Mathematical Sciences University of Delaware Newark, DE 19716, USA Email: rakesh@math.udel.edu January 14, 2011 1 Problem Formulation
More informationINCREASING STABILITY IN THE INVERSE PROBLEM FOR THE SCHRÖDINGER EQUATION. A Dissertation by. Li Liang
INCREASING STABILITY IN THE INVERSE PROBLEM FOR THE SCHRÖDINGER EQUATION A Dissertation by Li Liang Bachelor of Science, Wichita State University, 20 Master of Science, Wichita State University, 203 Submitted
More informationProceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 1115th July 2005
Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 1115th July 2005 SOME INVERSE SCATTERING PROBLEMS FOR TWODIMENSIONAL SCHRÖDINGER
More informationDETERMINING A FIRST ORDER PERTURBATION OF THE BIHARMONIC OPERATOR BY PARTIAL BOUNDARY MEASUREMENTS
DETERMINING A FIRST ORDER PERTURBATION OF THE BIHARMONIC OPERATOR BY PARTIAL BOUNDARY MEASUREMENTS KATSIARYNA KRUPCHYK, MATTI LASSAS, AND GUNTHER UHLMANN Abstract. We consider an operator 2 + A(x) D +
More informationAN INVERSE PROBLEM FOR THE WAVE EQUATION WITH A TIME DEPENDENT COEFFICIENT
AN INVERSE PROBLEM FOR THE WAVE EQUATION WITH A TIME DEPENDENT COEFFICIENT Rakesh Department of Mathematics University of Delaware Newark, DE 19716 A.G.Ramm Department of Mathematics Kansas State University
More informationInverse Electromagnetic Problems
Title: Name: Affil./Addr. 1: Inverse Electromagnetic Problems Gunther Uhlmann, Ting Zhou University of Washington and University of California Irvine /gunther@math.washington.edu Affil./Addr. 2: 340 Rowland
More informationHyperbolic inverse problems and exact controllability
Hyperbolic inverse problems and exact controllability Lauri Oksanen University College London An inverse initial source problem Let M R n be a compact domain with smooth strictly convex boundary, and let
More informationarxiv: v1 [math.ap] 11 Jun 2007
Inverse Conductivity Problem for a Parabolic Equation using a Carleman Estimate with One Observation arxiv:0706.1422v1 [math.ap 11 Jun 2007 November 15, 2018 Patricia Gaitan Laboratoire d Analyse, Topologie,
More informationAALBORG UNIVERSITY. The Calderón problem with partial data for less smooth conductivities. Kim Knudsen. Department of Mathematical Sciences
AALBORG UNIVERSITY The Calderón problem with partial data for less smooth conductivities by Kim Knudsen R200507 February 2005 Department of Mathematical Sciences Aalborg University Fredrik Bajers Vej
More informationStability and instability in inverse problems
Stability and instability in inverse problems Mikhail I. Isaev supervisor: Roman G. Novikov Centre de Mathématiques Appliquées, École Polytechnique November 27, 2013. Plan of the presentation The Gel fand
More informationINVERSE BOUNDARY VALUE PROBLEMS FOR THE PERTURBED POLYHARMONIC OPERATOR
INVERSE BOUNDARY VALUE PROBLEMS FOR THE PERTURBED POLYHARMONIC OPERATOR KATSIARYNA KRUPCHYK, MATTI LASSAS, AND GUNTHER UHLMANN Abstract. We show that a first order perturbation A(x) D + q(x) of the polyharmonic
More informationSome issues on Electrical Impedance Tomography with complex coefficient
Some issues on Electrical Impedance Tomography with complex coefficient Elisa Francini (Università di Firenze) in collaboration with Elena Beretta and Sergio Vessella E. Francini (Università di Firenze)
More informationOn inverse problems in secondary oil recovery
On inverse problems in secondary oil recovery Victor Isakov October 5, 2007 Department of M athematics and Statistics W ichita State U niversity W ichita, KS 67260 0033, U.S.A. e mail : Introduction victor.isakov@wichita.edu
More informationCalderón s inverse problem in 2D Electrical Impedance Tomography
Calderón s inverse problem in 2D Electrical Impedance Tomography Kari Astala (University of Helsinki) Joint work with: Matti Lassas, Lassi Päivärinta, Samuli Siltanen, Jennifer Mueller and Alan Perämäki
More informationSYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS. Massimo Grosi Filomena Pacella S. L. Yadava. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 21, 2003, 211 226 SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS Massimo Grosi Filomena Pacella S.
More informationis measured (n is the unit outward normal) at the boundary, to define the Dirichlet
SIAM J MATH ANAL Vol 45, No 5, pp 700 709 c 013 Society for Industrial and Applied Mathematics ON A CALDERÓN PROBLEM IN FREQUENCY DIFFERENTIAL ELECTRICAL IMPEDANCE TOMOGRAPHY SUNGWHAN KIM AND ALEXANDRU
More informationInverse Scattering with Partial data on Asymptotically Hyperbolic Manifolds
Inverse Scattering with Partial data on Asymptotically Hyperbolic Manifolds Raphael Hora UFSC rhora@mtm.ufsc.br 29/04/2014 Raphael Hora (UFSC) Inverse Scattering with Partial data on AH Manifolds 29/04/2014
More informationLOCALIZED POTENTIALS IN ELECTRICAL IMPEDANCE TOMOGRAPHY. Bastian Gebauer. (Communicated by Otmar Scherzer)
Inverse Problems and Imaging Volume 2, No. 2, 2008, 251 269 Web site: http://www.aimsciences.org LOCALIZED POTENTIALS IN ELECTRICAL IMPEDANCE TOMOGRAPHY astian Gebauer Institut für Mathematik Johannes
More informationi=1 α i. Given an mtimes continuously
1 Fundamentals 1.1 Classification and characteristics Let Ω R d, d N, d 2, be an open set and α = (α 1,, α d ) T N d 0, N 0 := N {0}, a multiindex with α := d i=1 α i. Given an mtimes continuously differentiable
More informationThe Helically Reduced Wave Equation as a Symmetric Positive System
Utah State University DigitalCommons@USU All Physics Faculty Publications Physics 2003 The Helically Reduced Wave Equation as a Symmetric Positive System Charles G. Torre Utah State University Follow this
More informationLECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES)
LECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES) RAYTCHO LAZAROV 1 Notations and Basic Functional Spaces Scalar function in R d, d 1 will be denoted by u,
More informationRecall that any inner product space V has an associated norm defined by
Hilbert Spaces Recall that any inner product space V has an associated norm defined by v = v v. Thus an inner product space can be viewed as a special kind of normed vector space. In particular every inner
More informationSecond Order Elliptic PDE
Second Order Elliptic PDE T. Muthukumar tmk@iitk.ac.in December 16, 2014 Contents 1 A Quick Introduction to PDE 1 2 Classification of Second Order PDE 3 3 Linear Second Order Elliptic Operators 4 4 Periodic
More informationUniqueness in determining refractive indices by formally determined farfield data
Applicable Analysis, 2015 Vol. 94, No. 6, 1259 1269, http://dx.doi.org/10.1080/00036811.2014.924215 Uniqueness in determining refractive indices by formally determined farfield data Guanghui Hu a, Jingzhi
More informationTraces, extensions and conormal derivatives for elliptic systems on Lipschitz domains
Traces, extensions and conormal derivatives for elliptic systems on Lipschitz domains Sergey E. Mikhailov Brunel University West London, Department of Mathematics, Uxbridge, UB8 3PH, UK J. Math. Analysis
More informationSELFADJOINTNESS OF SCHRÖDINGERTYPE OPERATORS WITH SINGULAR POTENTIALS ON MANIFOLDS OF BOUNDED GEOMETRY
Electronic Journal of Differential Equations, Vol. 2003(2003), No.??, pp. 1 8. ISSN: 10726691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) SELFADJOINTNESS
More informationThe double layer potential
The double layer potential In this project, our goal is to explain how the Dirichlet problem for a linear elliptic partial differential equation can be converted into an integral equation by representing
More informationShort note on compact operators  Monday 24 th March, Sylvester ErikssonBique
Short note on compact operators  Monday 24 th March, 2014 Sylvester ErikssonBique 1 Introduction In this note I will give a short outline about the structure theory of compact operators. I restrict attention
More informationReconstructing conductivities with boundary corrected Dbar method
Reconstructing conductivities with boundary corrected Dbar method Janne Tamminen June 24, 2011 Short introduction to EIT The Boundary correction procedure The Dbar method Simulation of measurement data,
More informationIncreasing stability in an inverse problem for the acoustic equation
Increasing stability in an inverse problem for the acoustic equation Sei Nagayasu Gunther Uhlmann JennNan Wang Abstract In this work we study the inverse boundary value problem of determining the refractive
More informationSHARP BOUNDARY TRACE INEQUALITIES. 1. Introduction
SHARP BOUNDARY TRACE INEQUALITIES GILES AUCHMUTY Abstract. This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region R N. The inequalities bound (semi)norms
More informationA local estimate from Radon transform and stability of Inverse EIT with partial data
A local estimate from Radon transform and stability of Inverse EIT with partial data Alberto Ruiz Universidad Autónoma de Madrid U. California, Irvine.June 2012 w/ P. Caro (U. Helsinki) and D. Dos Santos
More informationNumerical Harmonic Analysis on the Hyperbolic Plane
Numerical Harmonic Analysis on the Hyperbolic Plane Buma Fridman, Peter Kuchment, Kirk Lancaster, Serguei Lissianoi, Mila Mogilevsky, Daowei Ma, Igor Ponomarev, and Vassilis Papanicolaou Mathematics and
More informationPartial Differential Equations
Part II Partial Differential Equations Year 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2015 Paper 4, Section II 29E Partial Differential Equations 72 (a) Show that the Cauchy problem for u(x,
More informationA SHARP STABILITY ESTIMATE IN TENSOR TOMOGRAPHY
A SHARP STABILITY ESTIMATE IN TENSOR TOMOGRAPHY PLAMEN STEFANOV 1. Introduction Let (M, g) be a compact Riemannian manifold with boundary. The geodesic ray transform I of symmetric 2tensor fields f is
More informationSPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS
SPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS TSOGTGEREL GANTUMUR Abstract. After establishing discrete spectra for a large class of elliptic operators, we present some fundamental spectral properties
More informationNotions such as convergent sequence and Cauchy sequence make sense for any metric space. Convergent Sequences are Cauchy
Banach Spaces These notes provide an introduction to Banach spaces, which are complete normed vector spaces. For the purposes of these notes, all vector spaces are assumed to be over the real numbers.
More informationA Remark on the Regularity of Solutions of Maxwell s Equations on Lipschitz Domains
A Remark on the Regularity of Solutions of Maxwell s Equations on Lipschitz Domains Martin Costabel Abstract Let u be a vector field on a bounded Lipschitz domain in R 3, and let u together with its divergence
More informationNONLINEAR FREDHOLM ALTERNATIVE FOR THE plaplacian UNDER NONHOMOGENEOUS NEUMANN BOUNDARY CONDITION
Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 210, pp. 1 7. ISSN: 10726691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONLINEAR FREDHOLM ALTERNATIVE FOR THE plaplacian
More informationTitle: Localized selfadjointness of Schrödingertype operators on Riemannian manifolds. Proposed running head: Schrödingertype operators on
Title: Localized selfadjointness of Schrödingertype operators on Riemannian manifolds. Proposed running head: Schrödingertype operators on manifolds. Author: Ognjen Milatovic Department Address: Department
More informationTakens embedding theorem for infinitedimensional dynamical systems
Takens embedding theorem for infinitedimensional dynamical systems James C. Robinson Mathematics Institute, University of Warwick, Coventry, CV4 7AL, U.K. Email: jcr@maths.warwick.ac.uk Abstract. Takens
More informationFriedrich symmetric systems
viii CHAPTER 8 Friedrich symmetric systems In this chapter, we describe a theory due to Friedrich [13] for positive symmetric systems, which gives the existence and uniqueness of weak solutions of boundary
More informationRIEMANNHILBERT PROBLEM FOR THE MOISILTEODORESCU SYSTEM IN MULTIPLE CONNECTED DOMAINS
Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 310, pp. 1 5. ISSN: 10726691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu RIEMANNHILBERT PROBLEM FOR THE MOISILTEODORESCU
More informationarxiv: v3 [math.ap] 4 Jan 2017
Recovery of an embedded obstacle and its surrounding medium by formallydetermined scattering data Hongyu Liu 1 and Xiaodong Liu arxiv:1610.05836v3 [math.ap] 4 Jan 017 1 Department of Mathematics, Hong
More informationINVERSE BOUNDARY VALUE PROBLEMS FOR THE MAGNETIC SCHRÖDINGER EQUATION
INVERSE BOUNDARY VALUE PROBLEMS FOR THE MAGNETIC SCHRÖDINGER EQUATION MIKKO SALO Abstract. We survey recent results on inverse boundary value problems for the magnetic Schrödinger equation. 1. Introduction
More informationProblem Set 2: Solutions Math 201A: Fall 2016
Problem Set 2: s Math 201A: Fall 2016 Problem 1. (a) Prove that a closed subset of a complete metric space is complete. (b) Prove that a closed subset of a compact metric space is compact. (c) Prove that
More informationA GENERALIZATION OF THE FLAT CONE CONDITION FOR REGULARITY OF SOLUTIONS OF ELLIPTIC EQUATIONS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 100, Number 2. June 1987 A GENERALIZATION OF THE FLAT CONE CONDITION FOR REGULARITY OF SOLUTIONS OF ELLIPTIC EQUATIONS GARY M. LIEBERMAN ABSTRACT.
More informationMATH 220: MIDTERM OCTOBER 29, 2015
MATH 22: MIDTERM OCTOBER 29, 25 This is a closed book, closed notes, no electronic devices exam. There are 5 problems. Solve Problems 3 and one of Problems 4 and 5. Write your solutions to problems and
More informationCALDERÓN INVERSE PROBLEM FOR THE SCHRÖDINGER OPERATOR ON RIEMANN SURFACES
CALDERÓN INVERSE PROBLEM FOR THE SCHRÖDINGER OPERATOR ON RIEMANN SURFACES COLIN GUILLARMOU AND LEO TZOU Abstract. On a fixed smooth compact Riemann surface with boundary (M 0, g), we show that the Cauchy
More informationMAT 578 FUNCTIONAL ANALYSIS EXERCISES
MAT 578 FUNCTIONAL ANALYSIS EXERCISES JOHN QUIGG Exercise 1. Prove that if A is bounded in a topological vector space, then for every neighborhood V of 0 there exists c > 0 such that tv A for all t > c.
More informationFunctional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability...
Functional Analysis Franck Sueur 20182019 Contents 1 Metric spaces 1 1.1 Definitions........................................ 1 1.2 Completeness...................................... 3 1.3 Compactness......................................
More informationA note on W 1,p estimates for quasilinear parabolic equations
200Luminy conference on Quasilinear Elliptic and Parabolic Equations and Systems, Electronic Journal of Differential Equations, Conference 08, 2002, pp 2 3. http://ejde.math.swt.edu or http://ejde.math.unt.edu
More informationIntroduction to Real Analysis Alternative Chapter 1
Christopher Heil Introduction to Real Analysis Alternative Chapter 1 A Primer on Norms and Banach Spaces Last Updated: March 10, 2018 c 2018 by Christopher Heil Chapter 1 A Primer on Norms and Banach Spaces
More informationLECTURE 5 APPLICATIONS OF BDIE METHOD: ACOUSTIC SCATTERING BY INHOMOGENEOUS ANISOTROPIC OBSTACLES DAVID NATROSHVILI
LECTURE 5 APPLICATIONS OF BDIE METHOD: ACOUSTIC SCATTERING BY INHOMOGENEOUS ANISOTROPIC OBSTACLES DAVID NATROSHVILI Georgian Technical University Tbilisi, GEORGIA 00 1. Formulation of the corresponding
More informationANGKANA RÜLAND AND MIKKO SALO
QUANTITATIVE RUNGE APPROXIMATION AND INVERSE PROBLEMS ANGKANA RÜLAND AND MIKKO SALO Abstract. In this short note we provide a quantitative version of the classical Runge approximation property for second
More informationDiscreteness of Transmission Eigenvalues via Upper Triangular Compact Operators
Discreteness of Transmission Eigenvalues via Upper Triangular Compact Operators John Sylvester Department of Mathematics University of Washington Seattle, Washington 98195 U.S.A. June 3, 2011 This research
More informationChapter 1 Foundations of Elliptic Boundary Value Problems 1.1 Euler equations of variational problems
Chapter 1 Foundations of Elliptic Boundary Value Problems 1.1 Euler equations of variational problems Elliptic boundary value problems often occur as the Euler equations of variational problems the latter
More informationu xx + u yy = 0. (5.1)
Chapter 5 Laplace Equation The following equation is called Laplace equation in two independent variables x, y: The nonhomogeneous problem u xx + u yy =. (5.1) u xx + u yy = F, (5.) where F is a function
More informationA New Proof of AntiMaximum Principle Via A Bifurcation Approach
A New Proof of AntiMaximum Principle Via A Bifurcation Approach Junping Shi Abstract We use a bifurcation approach to prove an abstract version of antimaximum principle. The proof is different from previous
More informationWavelet moment method for Cauchy problem for the Helmholtz equation
Wavelet moment method for Cauchy problem for the Helmholtz equation Preliminary report Teresa Regińska, Andrzej Wakulicz December 9, 2005  Institute of Mathematics, Polish Academy of Sciences, email:
More informationIntegral Representation Formula, Boundary Integral Operators and Calderón projection
Integral Representation Formula, Boundary Integral Operators and Calderón projection Seminar BEM on Wave Scattering Franziska Weber ETH Zürich October 22, 2010 Outline Integral Representation Formula Newton
More informationPropagation of Smallness and the Uniqueness of Solutions to Some Elliptic Equations in the Plane
Journal of Mathematical Analysis and Applications 267, 460 470 (2002) doi:10.1006/jmaa.2001.7769, available online at http://www.idealibrary.com on Propagation of Smallness and the Uniqueness of Solutions
More informationNonuniqueness result for a hybrid inverse problem
Nonuniqueness result for a hybrid inverse problem Guillaume Bal and Kui Ren Abstract. Hybrid inverse problems aim to combine two imaging modalities, one that displays large contrast and one that gives
More informationReal Analysis Notes. Thomas Goller
Real Analysis Notes Thomas Goller September 4, 2011 Contents 1 Abstract Measure Spaces 2 1.1 Basic Definitions........................... 2 1.2 Measurable Functions........................ 2 1.3 Integration..............................
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Sobolev Embedding Theorems Embedding Operators and the Sobolev Embedding Theorem
More informationNonuniqueness result for a hybrid inverse problem
Nonuniqueness result for a hybrid inverse problem Guillaume Bal and Kui Ren Abstract. Hybrid inverse problems aim to combine two imaging modalities, one that displays large contrast and one that gives
More informationAllan Greenleaf, Matti Lassas, and Gunther Uhlmann
Mathematical Research Letters 10, 685 693 (2003) ON NONUNIQUENESS FOR CALDERÓN S INVERSE PROBLEM Allan Greenleaf, Matti Lassas, and Gunther Uhlmann Abstract. We construct anisotropic conductivities with
More informationON THE EXISTENCE OF TRANSMISSION EIGENVALUES. Andreas Kirsch1
Manuscript submitted to AIMS Journals Volume 3, Number 2, May 29 Website: http://aimsciences.org pp. 1 XX ON THE EXISTENCE OF TRANSMISSION EIGENVALUES Andreas Kirsch1 University of Karlsruhe epartment
More informationSYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS. M. Grossi S. Kesavan F. Pacella M. Ramaswamy. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 12, 1998, 47 59 SYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS M. Grossi S. Kesavan F. Pacella M. Ramaswamy
More informationSpectral theory of first order elliptic systems
Spectral theory of first order elliptic systems Dmitri Vassiliev (University College London) 24 May 2013 Conference Complex Analysis & Dynamical Systems VI Nahariya, Israel 1 Typical problem in my subject
More informationPOINTWISE BOUNDS ON QUASIMODES OF SEMICLASSICAL SCHRÖDINGER OPERATORS IN DIMENSION TWO
POINTWISE BOUNDS ON QUASIMODES OF SEMICLASSICAL SCHRÖDINGER OPERATORS IN DIMENSION TWO HART F. SMITH AND MACIEJ ZWORSKI Abstract. We prove optimal pointwise bounds on quasimodes of semiclassical Schrödinger
More informationThe Factorization Method for Inverse Scattering Problems Part I
The Factorization Method for Inverse Scattering Problems Part I Andreas Kirsch Madrid 2011 Department of Mathematics KIT University of the State of BadenWürttemberg and National Largescale Research Center
More informationA RELATIONSHIP BETWEEN THE DIRICHLET AND REGULARITY PROBLEMS FOR ELLIPTIC EQUATIONS. Zhongwei Shen
A RELATIONSHIP BETWEEN THE DIRICHLET AND REGULARITY PROBLEMS FOR ELLIPTIC EQUATIONS Zhongwei Shen Abstract. Let L = diva be a real, symmetric second order elliptic operator with bounded measurable coefficients.
More informationarxiv: v1 [math.ap] 16 Jan 2015
Three positive solutions of a nonlinear Dirichlet problem with competing power nonlinearities Vladimir Lubyshev January 19, 2015 arxiv:1501.03870v1 [math.ap] 16 Jan 2015 Abstract This paper studies a nonlinear
More informationStabilization and Controllability for the Transmission Wave Equation
1900 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO. 12, DECEMBER 2001 Stabilization Controllability for the Transmission Wave Equation Weijiu Liu Abstract In this paper, we address the problem of
More informationEigenvalues and eigenfunctions of the Laplacian. Andrew Hassell
Eigenvalues and eigenfunctions of the Laplacian Andrew Hassell 1 2 The setting In this talk I will consider the Laplace operator,, on various geometric spaces M. Here, M will be either a bounded Euclidean
More informationLECTURE 15: COMPLETENESS AND CONVEXITY
LECTURE 15: COMPLETENESS AND CONVEXITY 1. The HopfRinow Theorem Recall that a Riemannian manifold (M, g) is called geodesically complete if the maximal defining interval of any geodesic is R. On the other
More informationNondegeneracy of perturbed solutions of semilinear partial differential equations
Nondegeneracy of perturbed solutions of semilinear partial differential equations Robert Magnus, Olivier Moschetta Abstract The equation u + FV εx, u = 0 is considered in R n. For small ε > 0 it is shown
More informationLocally convex spaces, the hyperplane separation theorem, and the KreinMilman theorem
56 Chapter 7 Locally convex spaces, the hyperplane separation theorem, and the KreinMilman theorem Recall that C(X) is not a normed linear space when X is not compact. On the other hand we could use semi
More informationKreinRutman Theorem and the Principal Eigenvalue
Chapter 1 KreinRutman Theorem and the Principal Eigenvalue The KreinRutman theorem plays a very important role in nonlinear partial differential equations, as it provides the abstract basis for the proof
More informationASYMPTOTIC BEHAVIOR OF GENERALIZED EIGENFUNCTIONS IN NBODY SCATTERING
ASYMPTOTIC BEHAVIOR OF GENERALIZED EIGENFUNCTIONS IN NBODY SCATTERING ANDRAS VASY Abstract. In this paper an asymptotic expansion is proved for locally (at infinity) outgoing functions on asymptotically
More informationAn inverse scattering problem in random media
An inverse scattering problem in random media Pedro Caro Joint work with: Tapio Helin & Matti Lassas Computational and Analytic Problems in Spectral Theory June 8, 2016 Outline Introduction and motivation
More informationNondegeneracy of perturbed solutions of semilinear partial differential equations
Nondegeneracy of perturbed solutions of semilinear partial differential equations Robert Magnus, Olivier Moschetta Abstract The equation u + F(V (εx, u = 0 is considered in R n. For small ε > 0 it is
More informationAn Accurate FourierSpectral Solver for Variable Coefficient Elliptic Equations
An Accurate FourierSpectral Solver for Variable Coefficient Elliptic Equations Moshe Israeli Computer Science Department, TechnionIsrael Institute of Technology, Technion city, Haifa 32000, ISRAEL Alexander
More informationTheory of PDE Homework 2
Theory of PDE Homework 2 Adrienne Sands April 18, 2017 In the following exercises we assume the coefficients of the various PDE are smooth and satisfy the uniform ellipticity condition. R n is always an
More informationVariational Formulations
Chapter 2 Variational Formulations In this chapter we will derive a variational (or weak) formulation of the elliptic boundary value problem (1.4). We will discuss all fundamental theoretical results that
More informationUSING FUNCTIONAL ANALYSIS AND SOBOLEV SPACES TO SOLVE POISSON S EQUATION
USING FUNCTIONAL ANALYSIS AND SOBOLEV SPACES TO SOLVE POISSON S EQUATION YI WANG Abstract. We study Banach and Hilbert spaces with an eye towards defining weak solutions to elliptic PDE. Using LaxMilgram
More informationThe Xray transform for a nonabelian connection in two dimensions
The Xray transform for a nonabelian connection in two dimensions David Finch Department of Mathematics Oregon State University Corvallis, OR, 97331, USA Gunther Uhlmann Department of Mathematics University
More informationON THE CORRECT FORMULATION OF A MULTIDIMENSIONAL PROBLEM FOR STRICTLY HYPERBOLIC EQUATIONS OF HIGHER ORDER
Georgian Mathematical Journal 1(1994), No., 141150 ON THE CORRECT FORMULATION OF A MULTIDIMENSIONAL PROBLEM FOR STRICTLY HYPERBOLIC EQUATIONS OF HIGHER ORDER S. KHARIBEGASHVILI Abstract. A theorem of
More informationVelocity averaging a general framework
Outline Velocity averaging a general framework Martin Lazar BCAM ERCNUMERIWAVES Seminar May 15, 2013 Joint work with D. Mitrović, University of Montenegro, Montenegro Outline Outline 1 2 L p, p >= 2 setting
More informationSelfintersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds
Selfintersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds John Douglas Moore Department of Mathematics University of California Santa Barbara, CA, USA 93106 email: moore@math.ucsb.edu
More informationAN INVERSE PROBLEM OF CALDERÓN TYPE WITH PARTIAL DATA
AN INVERSE PROBLEM OF CALDERÓN TYPE WITH PARTIAL DATA JUSSI BEHRNDT AND JONATHAN ROHLEDER Abstract. A generalized variant of the Calderón problem from electrical impedance tomography with partial data
More informationInverse scattering problem with underdetermined data.
Math. Methods in Natur. Phenom. (MMNP), 9, N5, (2014), 244253. Inverse scattering problem with underdetermined data. A. G. Ramm Mathematics epartment, Kansas State University, Manhattan, KS 665062602,
More informationFIXED POINT METHODS IN NONLINEAR ANALYSIS
FIXED POINT METHODS IN NONLINEAR ANALYSIS ZACHARY SMITH Abstract. In this paper we present a selection of fixed point theorems with applications in nonlinear analysis. We begin with the Banach fixed point
More informationREGULARITY RESULTS FOR THE EQUATION u 11 u 22 = Introduction
REGULARITY RESULTS FOR THE EQUATION u 11 u 22 = 1 CONNOR MOONEY AND OVIDIU SAVIN Abstract. We study the equation u 11 u 22 = 1 in R 2. Our results include an interior C 2 estimate, classical solvability
More information