ON THE EXISTENCE OF TRANSMISSION EIGENVALUES. Andreas Kirsch1

Size: px
Start display at page:

Download "ON THE EXISTENCE OF TRANSMISSION EIGENVALUES. Andreas Kirsch1"

Transcription

1 Manuscript submitted to AIMS Journals Volume 3, Number 2, May 29 Website: pp. 1 XX ON THE EXISTENCE OF TRANSMISSION EIGENVALUES Andreas Kirsch1 University of Karlsruhe epartment of Mathematics Karlsruhe Germany Abstract. The investigation of the far field operator and the Factorization Method in inverse scattering theory leads naturally to the study of corresponding interior transmission eigenvalue problems. In contrast to the classical irichlet- or Neumann eigenvalue problem for in bounded domains these interior transmiision eigenvalue problem fail to be selfadjoint. In general, existence of eigenvalues is an open problem. In this paper we prove existence of eigenvalues for the scalar Helmholtz equation (isotropic and anisotropic cases) and Maxwell s equations under the condition that the contrast of the scattering medium is large enough. 1. Introduction. The relationship, among physicists sometimes called the insideoutside duality, between the eigenvalue one of the scattering matrix and the irichlet eigenvalues of the negative Laplacian is well known for a long time (see, e.g. 8] and the references therein). For this case the underlying scattering model is the exterior boundary value problem u s + k 2 u s in the exterior of, u s u inc on, for the Helmholtz equation. Here, denotes some bounded domain (the scatterer ), u inc the incident wave (a plane wave), and u s the scattered field which has to satisfy the Sommerfeld radiation condition. Note that the scattering problem is set up outside of while the eigenvalue problem is set up inside of. The analogous relationship for penetrable obstacles leads to new kind of eigenvalue problems in which are formulated as a pair of equations, coupled through the Cauchy data on the boundary. While the irichlet eigenvalue problem in is one of the best studied problems in analysis, the corresponding interior transmission eigenvalue problem is a relatively young object of research. We refer to the original papers 6, 12, 3, 1, 4, 5, 1, 9, 13, 14] and the monographs 2, 15] for the relevance of the transmission eigenvalue problems in acoustic and electromagnetic scattering theory. The recent survey paper 7] reports on the state of the art for the interior transmission problems till the end of 27. Since the eigenvalue problem doesn t seem to be treatable by standard methods in partial differential equations even some of the basic questions such as the existence of eigenvalues are still open up to today. This question of existence was raised for the first time in 6] and only last year (in 28) the problem has been partially answered by Lassi Päivärinta and John Sylvester 2 Mathematics Subject Classification. Primary: 35P15, 35J5; Secondary: 35P25, 35Q6. Key words and phrases. Helmholtz equation, Maxwell s equations, eigenvalue problem. 1

2 2 ANREAS KIRSCH in 17]. They prove existence of eigenvalues for the simplest model of a penetrable obstacle (in electromagnetics would this be the E-mode) provided the contrast is large enough. In this paper we will extend the analysis and treat the more complicated anisotropic cases, the H-mode in electromagnetics and the case of Maxwell s equations. The paper is organized as follows. In Section 2 we recall and, in our opinion, simplify the analysis of 17]. We adopt the notation of Section 4.5 of 15] and show discreteness of the spectrum and existence of eigenvalues, the latter under a condition on the contrast which is similar to the one in 17]. In Section 3 we consider the anisotropic case. Although the discreteness of the spectrum is well known (see 3, 2]) we suggest a different and more direct approach which models the approach for the first case. This approach makes it possible to derive a condition on the contrast such that eigenvalues exist. In Section 4 we show that the analysis carries over to the case of Maxwell s equations. 2. The Scalar Helmholtz Equation. We make the assumption that R 3 is some bounded connected domain with Lipschitz boundary. The two-dimensional case can be treated analogously. Furthermore, let q L () be real-valued such that q(x) q almost everywhere in for some q >. In the following all of the spaces consist of real-valued functions. This is not a restriction since otherwise one can pass over to the real- and imaginary parts. efinition 2.1. k 2 > is called an interior transmission eigenvalue if there exists real-valued (u, w) L 2 () L 2 () with (u, w) (, ) such that u w H 2 () and u + k 2 (1 + q)u in, w + k 2 w in, (2.1) u w on, u ν w on. (2.2) ν Here and in the following, ν ν(x) denotes the exterior unit normal vector for x and H 2 () denotes the Sobolev space of second order with vanishing traces v and v/ ν. We equip H 2 () with the inner product u, v H 2 () ( u), ( v)/q and corresponding norm L 2 () H 2() which is equivalent to the ordinary norm of H 2 () (see, e.g., 15]). We note that the traces exist in H 2 (). Therefore, the transmission boundary conditions (2.2) are already included in the space H 2 (). The differential equations (2.1) have to be understood in the ultra weak sense, i.e. w ψ + k 2 ψ ] dx for all ψ H 2 () and analogously for u. efining v w u we have the equivalent form that v H 2 () satisfies v + k 2 (1 + q)v k 2 q w in, (2.3) and w satisfies the Helmholtz equation in. To eliminate w from (2.3) we devide by q and apply the Helmholtz operator again. This yields ( + k 2 ) 1 ( + k 2 (1 + q) ) v, q

3 EXISTENCE OF TRANSMISSION EIGENVALUES 3 i.e. in weak form v + k 2 (1 + q)v ] ψ + k 2 ψ ] dx q for all ψ H 2 (). (2.4) We define the bilinear form a k for k by (2.4), i.e. a k (v, ψ) v + k 2 (1 + q)v ] ψ + k 2 ψ ] dx (2.5) q v + k 2 v ] ψ + k 2 ψ ] dx + k 2 v ψ + k 2 ψ ] dx(2.6) q for all ψ H 2 (). Then k 2 is an interior transmission eigenvalue if, and only if, there exists a non-trivial v H 2 () with a k (v, ψ) for all ψ H 2 (). Indeed, if k 2 is an interior transmission eigenvalue with corresponding eigenpair (u, w) L 2 () L 2 () then v w u H 2 () solves a k (v, ψ) for all ψ H 2 () as we have just seen. If, on the other hand, v H 2 () solves a k (v, ψ) for all ψ H 2 () then w 1 v + k 2 k 2 (1 + q)v ] q belongs to L 2 () and satisfies w ψ + k 2 ψ ] dx for all ψ H 2 (), which is the ultra weak form of w + k 2 w in. We write a k in the form a k a + k 2 b 1 + k 4 b 2 where the bilinear forms b 1 and b 2 are given by ] dx b 1 (v, ψ) v ψ + ψ v + v ψ dx, q q + 1 b 2 (v, ψ) v ψ dx, v, ψ H 2 (), q and a (, ) is just the inner product in H 2 (). By the representation theorem of Riesz there exist bounded operators B 1, B 2 from H 2 () into itself with b j (v, ψ) B j v, ψ H 2 () for all v, ψ H 2 (), j 1, 2. The equation a k (v, ψ) for all ψ H 2 () takes the form v + k 2 B 1 v + k 4 B 2 v. (2.7) Since b j are symmetric (easy to see by Green s second formula!) we observe that B j are self adjoint for j 1, 2. Also, as shown in 15] the operators B j are compact and B 2 is positive. Therefore, the operator B 2 has a positive square root B 1/2 2 : H 2 () H 2 (). Setting z k 2 B 1/2 2 v we observe that (2.7) is equivalent to the system ( ) v z + k 2 ( B 1 B 1/2 2 B 1/2 2 ) (v ) z ( ). (2.8)

4 4 ANREAS KIRSCH This is a non self adjoint linear eigenvalue problem for a compact matrix operator. We conclude that the spectrum is discrete but we cannot conclude existence of any eigenvalues. We set A k Id + k 2 B 1 + k 4 B 2 and note that the spectrum of A k is real and discrete with one as the only accumulation point. The operator and therefore also the eigenvalues depend continuously on k (see, e.g., 11]). Since A Id the spectrum σ(a ) consists of 1 only. The following arguments for showing existence of transmission eigenvalues have been recently suggested by Päivärinta and Sylvester in 17]. The idea is to construct some ˆv H 2 () and some ˆk > such that Aˆkˆv, ˆv H 2 () aˆk(ˆv, ˆv) <. By the min-max principle this implies that the smallest eigenvalue of Aˆk is negative. Therefore, since the smallest eigenvalue depends continuously on k, there exists k between and ˆk such that A k has zero as the smallest eigenvalue i.e., in particular, possesses an eigenvalue at all. To carry out this idea we estimate a k (v, v) from above. From (2.6) we have a k (v, v) 1 v + k 2 v ] 2 dx + k 2 v v dx + k 4 v 2 L q 2 () 1 ( v) 2 + k 2 (2 + q ) v v ] dx + (1 + q )k 4 q 1 ( v) 2 k 2 (2 + q ) v 2] dx + (1 + q )k 4 q q q v 2 L 2 () v 2 L 2 () where we applied Green s first theorem in the last step. Let now ˆv be an eigenfunction corresponding to the smallest eigenvalue µ 1 of the bi-laplacian 2, i.e. ˆv H 2 () satisfies 2ˆv µ 1ˆv in. Green s second theorem yields ( ˆv)2 dx µ 1 ˆv2 dx and thus a k (ˆv, ˆv) µ 1 + k 4 (1 + q ) q ˆv 2 L 2 () k2 (2 + q ) q ˆv 2 L 2 (). Now let ρ be the smallest irichlet eigenvalue of in. Then Poincaré s inequality yields u 2 L 2 () ρ 1 u 2 L 2 () for all u H1 () and thus, since H 2 () H 1 (), a k (ˆv, ˆv) 1 ρ q µ1 + k 4 (1 + q ) k 2 ρ (2 + q ) ] ˆv 2 L 2 (). We can now easily derive a condition on q such the term on the right hand side is negative. First we write (completing the square) ( µ 1 +k 4 (1+q ) k 2 ρ (2+q ) k q (1 + q ) 2 /2)ρ +µ 1 (1 + q /2) 2 ρ q 1 + q We choose k 2 such that the square vanishes. Then the expression is negative if µ 1 < (1 + q /2) 2 ρ q which can be rewritten as (note that µ 1 ρ 2 by, e.g., 17]) ( ) ] µ1 µ1 µ1 q > (2.9) ρ 2 ρ ρ 2

5 EXISTENCE OF TRANSMISSION EIGENVALUES 5 Therefore, for this particular choice of ˆk 2 (1 + q /2)ρ /(1 + q ) we have inf aˆk(v, v) <. v H 2() Therefore, the smallest eigenvalue of Aˆk must be negative. Since the spectrum of A is positive there must be some k between and ˆk such that the smallest eigenvalue of A k is zero. This k 2 is a transmission eigenvalue! We summarize the result in the following theorem. Theorem 2.2. Let ρ > be the smallest irichlet eigenvalue of in and µ 1 > be the smallest eigenvalue of 2 with respect to the boundary conditions v on and v/ ν on. Assume that q satisfies (2.9). Then there exists at least one transmission eigenvalue k 2. By the same arguments as in 17] one can extend this to prove existence of at least m eigenvalues. The basis is set by the following theorem. Let V m be the set of linear subspaces of H 2 () of co-dimension m. Theorem 2.3. For any m 1, 2,... define f m : R R by f m (k) sup inf V V m 1 v V, v A k v, v H 2 () v 2 H 2 (), k. (2.1) (a) f m is continuous on, ) with f m () 1. (b) If f m (k) < 1 for some k > then there exist m eigenvalues λ j λ j (k), j 1,..., m, of A k less than 1 (counted according to their multiplicities), ordered as λ 1 λ 2 λ m < 1 and f m (k) λ m. In this case the supremum is attained for the subspace V span{v 1,..., v m 1 } where v j H 2 () are the eigenfunctions corresponding to the eigenvalues λ j. We do not prove this result but refer to, e.g., 18]. Let now µ 1 µ 2 µ m be the m smallest eigenvalues of 2 with respect to homogeneous boundary conditions v on and v/ ν on with corresponding eigenfunctions ˆv 1,..., ˆv m H 2 (). If we choose q according to (2.9) where we replace µ 1 by µ m then A k v, v H 2 () < for all v span{ˆv 1,..., ˆv m }. Since V span{ˆv 1,..., ˆv m } for every V V m 1 we conclude that f m (ˆk) where ˆk is again given by ˆk 2 (1 + q /2)ρ /(1 + q ). Therefore, the previous theorem is applicable which yields f m (ˆk) λ m (ˆk). By the continuity of f m there exists a largest k m (, ˆk] with f m (k m ). Therefore, k m is a transmission eigenvalue. Let d m {1,..., m} be the multiplicity of the eigenvalue zero of A km. If d m < m then f m dm (k m ) <. Therefore, there exists k m dm (, k m ) with f m dm (k m dm ). In this way we proceed and arrive at the following corollary. Corollary 2.4. Let µ j, j 1, 2, 3,..., be the eigenvalues of 2 with respect to homogeneous boundary conditions as in Theorem 2.2. We assume that they are ordered as µ 1 µ 2 and they appear according to their multiplicity. if q satisfies ( ) ] µm µm µm q > 2 ρ ρ ρ 2 1 then there exist at least m transmission eigenvalues (counted according to their multiplicities).

6 6 ANREAS KIRSCH 3. The Anisotropic Case. We make the assumption that Q L (, R 3 3 ) is matrix-valued such that Q(x) is real and symmetric for almost all x R 3. Furthermore, we assume that there exists q > such that z Q(x)z q z 2 for all z R 3 almost everywhere on. Again, is a bounded and connected domain with Lipschitz boundary. Acoustic scattering with space dependent density leads to the following interior transmission eigenvalue problem (cf. 3, 2, 14]). efinition 3.1. k 2 > is called an interior transmission eigenvalue if there exists real-valued (u, w) H 1 () H 1 () with (u, w) (, ) such that w + k 2 w in and div ( (I + Q) u ) + k 2 u in and the Cauchy data of u and v coincide, i.e. u w on and ν (I + Q) u w/ ν on. The variational forms are w ψ k 2 w ψ ] dx for all ψ H 1 (), (3.11) u (I + Q) ψ k 2 u ψ ] dx w ψ k 2 w ψ ] dx (3.12) for all ψ H 1 (). If (u, w) solves (3.11), (3.12) with u w on then v : w u H 1 () solves v (I +Q) ψ k 2 vψ ] dx w Q ψ dx for all ψ H 1 (). (3.13) Note that ψ 1 yields (provided k ) that vdx, i.e. { } v H 1 () : v H 1 () : vdx. Analogously, we define the space H 1 () as the subspace of H 1 () of functions with vanishing means. The classical form of (3.13) is div ( (I + Q) v ) + k 2 v div(q w) in, ν (I + Q) v ν Q w on, (3.14) and v on. The idea of Section 2 to eliminate w explicitely does not work here. Howewer, we can express w implicitely by v. We carry out the details and define the operator L k from H 1 () into itself as follows. For given v H 1 () let w w v H 1 () be the unique solution of the Neumann problem (3.14), i.e. w Q ψ dx v (I +Q) ψ k 2 vψ ] dx for all ψ H 1 (). (3.15) Note that the solution w w v H 1 () exists and is unique because of vdx. Let z z v H 1 () be the unique representation of the linear and bounded functional ψ w v ψ k 2 w v ψ ] dx, ψ H 1 (), (3.16) i.e. z v, ψ H 1 () w v ψ k 2 w v ψ ] dx for all ψ H 1 ().

7 EXISTENCE OF TRANSMISSION EIGENVALUES 7 Then we set L k v z v. Theorem 3.2. (a) Let (u, w) H 1 () H 1 () be an eigenfunction. Then v w u H 1 () solves L k v. (b) Let v H 1 () satisfy L k v. Furthermore, let w w v H 1 () be as in the construction of L k, i.e. the solution of (3.15). Then there exists a constant c R such that (u, w + c) is an eigenfunction where u w + c v. Proof. (a) Formula (3.11) implies z v, ψ H 1 () for all ψ H 1 (), i.e. L k v z v. (b) Let, on the other hand, L k v, i.e. w ψ k 2 wψ ] dx for all ψ H 1 (). Here and in the following we write w for w v. Note that this does not imply that w solves the Helmholtz equation because of the restriction ψdx on the test functions. However, fix a function φ H 1 () with φ dx 1. Let ψ H1 () be any function. Then ψ ψ ( ψ dx) φ H 1 () and thus w ψ k 2 w ψ ] dx w ψ k 2 wψ ] dx ψdx (w + c) ψ k 2 (w + c)ψ ] dx. w φ k 2 wφ ] dx } {{ } : k 2 c This shows that w + c solves the Helmholtz equation in. We set u : w + c v and observe that the Cauchy data of w + c and u coincide. Furthermore, equation (3.12) follows from (3.15). Therefore, the transmission eigenvalues are just the parameters k 2 for which L k fails to be injective. The operator L k has the same form as the corresponding operator of equation (2.7) as we see from the next theorem. Theorem 3.3. (a) L k has the form L k L + k 2 C 1 + k 4 C 2 with self adjoint compact operators C j from H 1 () into itself. (b) L is self adjoint and coercive on H 1 (), in particular L v, v H 1 () v 2 L 2 () c v 2 H 1 () for all v H 1 () where c > is independent of v. Proof. (a) From the definition of w w v H 1 () we observe that w has the form w w 1 k 2 w 2 where w 1, w 2 H 1 () solve w1 Q ψ dx v (I + Q) ψ dx for all ψ H 1 (), w 2 Q ψ dx vψ dx for all ψ H 1 ().

8 8 ANREAS KIRSCH Substituting w w 1 k 2 w 2 into the form of the functional (3.16) yields the form L k L + k 2 C 1 + k 4 C 2. We show that L k is symmetric for every k. Then also C 1 and C 2 are symmetric as the first and second derivative, respectively, of L k with respect to k 2 at zero. For v 1, v 2 H 1 () we conclude L k v 1, v 2 H 1 () w 1 v 2 k 2 ] w 1 v 2 dx w 1 (I + Q) v 2 k 2 ] w 1 v 2 dx w1 Q v 2 dx where w j w vj. Now we use (3.15) twice: First for v v 2, ψ w 1, then for w w 1, ψ v 2. This yields L k v 1, v 2 H 1 () w2 Q w 1 dx v 1 (I+Q) v 2 k 2 ] v 1 v 2 dx, (3.17) and this is a symmetric expression in v 1 and v 2. The compactness of the operators C 1 and C 2 is easily seen by the compactness of the imbedding of H 1 () in L 2 (). We omit this proof but carry out the corresponding proof for the slightly more complicated electromagnetic case in Theorem 4.4 below. (b) For k and v 1 v 2 v equation (3.17) reduces to L v, v H 1 () w Q w dx v (I + Q) v dx. (3.18) Now we make use of the fact that, for almost all x, there exists a unique positive definite matrix Q 1/2 L (, R 3 3 ) with Q 1/2 Q 1/2 Q. From (3.15) for ψ v and k we estimate v (I + Q) v dx w Q v dx Q 1/2 w L 2 () Q 1/2 v L 2 () and thus L v, v H 1 () Q 1/2 w 2 L 2 () Q1/2 w L 2 () Q 1/2 v L 2 () Q 1/2 w L () 2 Q 1/2 w L 2 () Q 1/2 ] v L 2 () (3.19) and thus Q 1/2 w L 2 () Q 1/2 w L 2 () + Q 1/2 v L 2 () }{{} :c(v) Q 1/2 w 2 L 2 () Q1/2 v 2 ] L 2 () c(v) w Q w v (I + Q) v ] dx } {{ } L v,v H 1 () ( 1 c(v) ) L v, v H 1 () c(v) v 2 L 2 (). + c(v) v 2 dx

9 EXISTENCE OF TRANSMISSION EIGENVALUES 9 We note that c(v) < 1 for v, thus L v, v H 1 () c(v) 1 c(v) v 2 L 2 (). We show that c(v) 1 2. From (3.19) we conclude that Q 1/2 v 2 L 2 () v Q v dx v (I + Q) v dx Q 1/2 w L 2 () Q 1/2 v L 2 (), i.e. Q 1/2 v L 2 () Q 1/2 w L 2 (), i.e. c(v) 1 2. Finally, we note that v v L 2 () is an equivalent norm on H 1 () by Poincaré s inequality which proves the theorem. We write the equation L k v equation in the form and ṽ + k 2 L 1/2 C 1 L 1/2 ṽ + k 4 L 1/2 C 2 L 1/2 ṽ ṽ + k 2 B 1 ṽ + k 4 B 2 ṽ with obvious settings of B 1 and B 2. Here, L 1/2 denotes the (coercive) square root of the coercive operator L and ṽ L 1/2 v. As in (2.8) of the previous section we rewrite this as a linear eigenvalue system with a compact matrix operator and conclude the following theorem: Theorem 3.4. There exists at most a countable set of transmission eigenvalues, and the only possible accumulation point is infinity. Again, we want to show existence of some k > and some v such that L k v. Since the spectrum of Id + k 2 B 1 + k 4 B 2 for k is just {1} we follow again the idea of the previous section and show that, for sufficiently large values of q, there exists ˆk > and ˆv such that Lˆkˆv, ˆv H 1 (). Then, (L 1/2 ˆv) + ˆk 2 B 1 (L 1/2 + ˆk 4 B 2ˆv), (L 1/2 ˆv) H 1 (). Therefore, by continuity of the smallest eigenvalue of Id+k 2 B 1 +k 4 B 2 with respect to k there exists k between and ˆk such that the smallest eigenvalue of Id+k 2 B 1 + k 4 B 2 is zero. To carry out this idea we have to estimate L k v, v H 1 () from above. We have from (3.17) for v 1 v 2 v L k v, v H 1 () w Q w dx v (I + Q) v k 2 v 2] dx. We set again u w v and note that u solves the Neumann problem (in the weak sense) div Q u ] v + k 2 v in, ν Q u v on, (3.2) ν

10 1 ANREAS KIRSCH and thus L k v, v H 1 () ( v + u) Q( v + u) dx v (I + Q) v k 2 v 2] dx v 2 L 2 () + 2 Q v, u L 2 () + Q1/2 u 2 L 2 () + + k 2 v 2 L 2 (). From the weak form of (3.2) we conclude that and Q v, u L 2 () v 2 L 2 () k2 v 2 L 2 (), L k v, v H 1 () v 2 L 2 () k2 v 2 L 2 () + Q1/2 u 2 L 2 (). We estimate the last term. Again, from the weak form of (3.2) we have that Q u 2 dx v u k 2 vu ] dx v L 2 () u L 2 () + k 2 v L 2 () u L 2 (). Let ρ > be such that ψ L 2 () ρ ψ L 2 () for all ψ H 1 (). Then Q u 2 dx u L ()( 2 v L 2 () + k 2 ) ρ v L 2 () 1 q Q 1/2 u L 2 ()( v L 2 () + k 2 ρ v L 2 ()), i.e. after division of Q 1/2 u L 2 () and squaring, Q u 2 dx 1 ( v L q 2 () + k 2 ) 2 ρ v L 2 () Altogether we have the estimate 2 q ( v 2 L 2 () + k4 ρ 2 v 2 L 2 ()). L k v, v H 1 () v 2 L 2 () k2 v 2 L 2 () + 2 v 2 L q 2 () + 2k4 ρ2 v 2 L q 2 () ) ( (1 + 2q v 2L2() k2 1 2k2 ρ 2 ) v 2 L q 2 (). Now let µ be an eigenvalue with corresponding eigenfunction ˆv H 1 () of the eigenvalue problem ] v ψ µvψ dx for all ψ H1 (). (3.21) It is easily seen that such eigenvalues µ j exist, that is not an eigenvalue and that they are real and positive and converge to infinity. Taking ψ ˆv we conclude that ˆv 2 dx µ ˆv 2 dx.

11 EXISTENCE OF TRANSMISSION EIGENVALUES 11 Substituting v ˆv in the previous estimate we conclude that ) L kˆv, ˆv H 1 () (1 + 2q µ k 2 + 2k4 ρ 2 ] ˆv 2 L q 2 (). (3.22) Now we derive a condition on q such that ( L kˆv, ˆv ) for some k. Indeed, H 1 () we multiply the bracket ] by q and rewrite this as q ] (q + 2)µ + 2k 4 ρ 2 k 2 q 2 k 2 ρ q ] 2 ] 2 + (q + 2)µ q2 2 ρ 8 ρ 2. Now we determine q large enough such that the second bracket is non-positive, i.e. (q + 2)µ q2 8 ρ 2. (3.23) Then we choose k such that the first bracket vanishes, i.e. Then we have that Lˆkˆv, ˆv H 1 (). ˆk 2 q 4 ρ 2. We summarize the result in the following theorem. Theorem 3.5. Let ρ > be a constant such that an estimate of Poincaré s type holds for the space H 1 (), i.e. ψ L 2 () ρ ψ L 2 () for all ψ H 1 (). Furthermore, let µ > be some eigenvalue of the eigenvalue problem (3.21). Assume that q satisfies (3.23). Then there exists at least one transmission eigenvalue k 2. Again, we can extend this to prove existence of at least m eigenvalues: Corollary 3.6. Let µ j, j 1, 2, 3,..., be the eigenvalues of (3.21). We assume that they are ordered as µ 1 µ 2 and they appear according to their multiplicity. If q satisfies (q + 2)µ m q2 8 ρ 2 then there exist at least m transmission eigenvalues (counted according to their multiplicities). Remark 1. The classical formulation of the eigenvalue problem (3.21) is: Find v H 1 () such that v + µv const in, v on, v dx. 4. Maxwell s Equations. We make again the assumption that Q L (, R 3 3 ) is matrix-valued such that Q(x) is real and symmetric for almost all x. Furthermore, we assume that there exists q > such that z Q(x)z q z 2 for all z R 3 almost everywhere on. Again, is a bounded and connected domain with Lipschitz boundary. We consider scattering of time-harmonic electromagnetic waves in non-magnetic materials. We assume that the reader is familiar with the standard spaces in this context. The space H(curl, ) is defined as the completion of C () 3 with respect to the norm u H(curl,) u, u H(curl,)

12 12 ANREAS KIRSCH where u, v H(curl,) curl u curl v + u v ] dx. The subspace of vanishing tangential traces is denoted by H (curl, ), i.e. H (curl, ) { u H(curl, ) : ν u on }. The trace is well defined, see e.g. 16]. The study of the Factorization Method or already the question of uniqueness of the far field operator leads to the following interior transmission eigenvalue problem (see 5, 9, 13, 15]). efinition 4.1. k 2 > is called an interior transmission eigenvalue if there exists real-valued (u, w) H(curl, ) H(curl, ) with (u, w) (, ) such that curl 2 w k 2 w in and curl ( (I + Q) curl u ) k 2 u in and the Cauchy data of u and v coincide, i.e. ν u ν w on and ν ( (I + Q) curl u ) ν curl w on. The variational forms are curl w curl ψ k 2 w ψ ] dx for all ψ H (curl, ), (4.24) curl u (I + Q) curl ψ k 2 u ψ ] dx curl w curl ψ k 2 w ψ ] dx(4.25) for all ψ H(curl, ). We define again the difference v w u and observe that v H (curl, ) satisfies the equation curl ( (I + Q) curl v ) k 2 v curl(q curl w) in, ν ( (I + Q) curl v ) ν (Q curl w) on, i.e. in variational form curl v (I + Q) curl ψ k 2 v ψ ] dx curl w Q curl ψ dx (4.26) for all ψ H(curl, ). By setting ψ ρ for some ρ H 1 () we note from this equation that v V where V v H (curl, ) : v ρ dx for all ρ H 1 () is the space of H(curl, ) functions with vanishing normal and tangential traces which are divergence free. Indeed, for smooth functions the integral can be written, using the divergence theorem, as v ρ dx ρ divv dx ρ ν v ds. If this vanishes for all ρ H 1 () then the divergence of v vanishes in and the normal component of v vanishes on. Analogously, we define W w H(curl, ) : w ρ dx for all ρ H 1 ().

13 EXISTENCE OF TRANSMISSION EIGENVALUES 13 It will later be necessary to know the orthogonal complement of V in H (curl, ). For any ϕ H 1 () let v ϕ H (curl, ) be the Riesz representation of the functional ψ ϕ ψ dx, i.e. v ϕ, ψ H(curl,) ϕ ψ dx for all ψ H (curl, ), ϕ H 1 (). Lemma 4.2. Let V H (curl, ) be defined above. Then V is a closed subspace with orthogonal complement V { v ϕ H (curl, ) : ϕ H 1 () }. Proof. It is obvious that V is a closed subspace of H (curl, ). Let Ṽ be the space on the right hand side of the characterization of V. Then Ṽ and V are orthogonal to each other. Indeed, for v V and v ϕ Ṽ we have that (take ψ v in the definition of v ϕ ): v ϕ, v H(curl,) ϕ v dx and this vanishes by the definition of V. Let now v H (curl, ) be orthogonal to Ṽ. Then v, v ϕ H(curl,) ϕ v dx for all ϕ H 1 (). Therefore, v V. This ends the proof of the lemma. Now we define the operator L k from V into itself in the same way as in the previous section. First we observe that the bilinear form p(w, ψ) curl w Q curl ψ dx, w, ψ W, is coercive on W. This follows again from Corollary 3.51 of 16]. Therefore, for every v V there exists a unique w w v W such that curl wv Q curl ψ dx curl v (I + Q) curl ψ k 2 v ψ ] dx (4.27) for all ψ W. Again, let z z v V be the unique representation of the linear and bounded functional ψ curl w v curl ψ k 2 wv ψ ] dx, ψ V, (4.28) i.e. z v, ψ H(curl,) curl w v curl ψ k 2 wv ψ ] dx for all ψ V. Then we set L k v z v. Analogously to Theorem 3.2 we can show: Theorem 4.3. (a) Let (u, w) be a transmission eigenfunction corresponding to k. Then v w u V solves L k v.

14 14 ANREAS KIRSCH (b) Let v V satisfy L k v. Furthermore, let w w v W be as in the construction of L k, i.e. the solution of (4.27). Then there exists ϕ H 1 () such that ( u, w + ϕ ) is an eigenfunction where u w + ϕ v. Proof. Part (a) has been shown during the derivation of the operator L k. (b) Let v V such that L k v, i.e. curl w curl ψ k 2 w ψ ] dx for all ψ V, (4.29) where w w v W is determined from (4.27). Let ẑ H (curl, ) be the Riesz representation of the functional ψ curl w curl ψ k 2 w ψ ] dx on the space H (curl, ), i.e. ẑ, ψ H(curl,) curl w curl ψ k 2 w ψ ] dx for all ψ H (curl, ). Equation (4.29) implies that ẑ V. From Lemma 4.2 we conclude that there exists ϕ H 1 () such that ẑ v ϕ, i.e. curl w curl ψ k 2 w ψ ] dx ẑ, ψ H(curl,) v ϕ, ψ H(curl,) i.e. ϕ ψ dx for all ψ H (curl, ), curl w+ ϕ/k 2 ] curl ψ k 2 w+ ϕ/k 2] ψ ] dx for all ψ H (curl, ). This shows that w : w + ϕ/k 2 satisfies (4.24). Furthermore, from the definition of w we conclude that w and v satisfy (4.26) for all ψ W. It remains to show (4.26) for all ψ H(curl, ). Therefore, let ψ H(curl, ). By the classical Helmholtz decomposition there exists ψ W and ρ H 1 () such that ψ ψ + ρ. Since (4.26) holds for test functions of the form ρ trivially (note that v V!) we conclude that w and v satisfy (4.26) for all ψ H(curl, ) which is equivalent to (4.25) for w and u : w v. Again, the transmission eigenvalues are just the parameters k 2 for which L k fails to be injective. Now we continue with the investigation of L k. Analogously to Theorem 3.3 one can show: Theorem 4.4. (a) L k has the form L k L + k 2 C 1 + k 4 C 2 with self adjoint compact operators C j from V into itself. (b) L is self adjoint and coercive on V, in particular L v, v H(curl,) curl v 2 L 2 () c v 2 H(curl,) for all v V where c > is independent of v.

15 EXISTENCE OF TRANSMISSION EIGENVALUES 15 Proof. We write w w v in the form w w 1 k 2 w 2 where w 1, w 2 W satisfy curl w1 Q curl ψ dx curl v (I + Q) curl ψ dx, ψ W, (4.3) curl w 2 Q curl ψ dx v ψ dx, ψ W. (4.31) Then z z v has the form z z + k 2 z 1 + k 4 z 2 where z, z 1, z 2 V satisfy z, ψ H(curl,) curl w1 curl ψ dx, ψ V, z 1, ψ H(curl,) curl w 2 curl ψ + w1 ψ ] dx, ψ V, z 2, ψ H(curl,) w 2 ψ dx, ψ V. We have to show that v z 1 and v z 2 are compact in V. We show this only for v z 1. First we estimate i.e. z 1 2 H(curl,) curl w 2 L 2 () curl z 1 L 2 () + w 1 L 2 () z 1 L 2 () z 1 H(curl,) curl w2 L 2 () + w 1 L 2 ()], z 1 H(curl,) curl w 2 L 2 () + w 1 L 2 (). From (4.31) we conclude for ψ w 2 curl w 2 2 L 2 () 1 q Q 1/2 curl w 2 2 L 2 () 1 q v, w 2 L 2 () 1 q v L 2 () w 2 L 2 () i.e. z 1 H(curl,) 1 v L 2 () w 2 L 2 () + w 1 L 2 (). q Let now (v j ) converge weakly to zero in V. enote the corresponding functions w 1 and w 2 by w 1,j and w 2,j, respectively. Since the solution operators of (4.3) and (4.31) are bounded they converge weakly to zero in W. Now we use that W is compactly imbedded in L 2 () 3. We refer to Corollary 3.51 of 16] for a proof. Therefore, w 1,j L 2 () and w 2,j L 2 () converge to zero which implies z 1,j H(curl,), and the compactness of C 1 has been shown. The self adjointness of L, C 1, and C 2 as well as the coercivity of L are shown in the same way as in the proof of Theorem 3.3. The constant c in part (b) exists because the norm curl ψ L 2 () is equivalent to the norm ψ H(curl,) in the subspace V. Again, we write the equation L k v in the form i.e. ṽ + k 2 L 1/2 C 1 L 1/2 ṽ + k 4 L 1/2 C 2 L 1/2 ṽ ṽ + k 2 B 1 ṽ + k 4 B 2 ṽ

16 16 ANREAS KIRSCH with obvious settings of B 1 and B 2. As in (2.8) of Section 2 we rewrite this as a linear eigenvalue system with a compact matrix operator and conclude the following theorem: Theorem 4.5. There exists at most a countable set of transmission eigenvalues, and the only possible accumulation point is infinity. Now we continue with the proof that for sufficienly large q > eigenvalues indeed exist. We follow the same lines as in the previous section and write L k v, v H(curl,) z k, v H(curl,) curl w curl v k 2 w v ] dx curl w (I + Q) curl v k 2 w v ] dx curl w Q curl v dx curl w Q curl w dx curl v (I + Q) curl v k 2 v 2] dx where we have used (4.27) twice (for ψ w and for ψ v). Here, w W is the unique solution of (4.27). We set again u w v and note that u solves the Neumann problem (in the weak sense) curl Q curl u ] curl 2 v k 2 v in, ν Q curl u ν curl v on, (4.32) and thus L k v, v H(curl,) From the weak form of (4.32) we conclude that i.e. (curl v + curl u) Q(curl v + curl u) dx curl v (I + Q) curl v k 2 v 2] dx curl v 2 L 2 () + 2 Q curl v, curl u L 2 () + Q 1/2 curl u 2 L 2 () + k2 v 2 L 2 (). Q curl v, curl u L 2 () curl v 2 L 2 () k2 v 2 L 2 (), L k v, v H(curl,) curl v 2 L 2 () k2 v 2 L 2 () + Q1/2 curl u 2 L 2 (). We estimate again the last term. From the weak form of (4.32) we have Q 1/2 curl u 2 L 2 () curl v, curl u L 2 () k 2 v, u L 2 () curl v L 2 () curl u L 2 () + k 2 v L 2 () u L 2 () curl u L ()( 2 curl v L 2 () + k 2 ) ρ v L 2 ()

17 EXISTENCE OF TRANSMISSION EIGENVALUES 17 where ρ > is such that ψ L 2 () ρ curl ψ L 2 () for all ψ W. Such a constant exists by Corollary 3.51 of 16]. Therefore, Q 1/2 curl u 2 L 2 () 1 Q 1/2 curl u L ()( 2 curl v L 2 () + k 2 ρ v L ()) 2, q and thus Altogether we have Q 1/2 curl u 2 L 2 () 2 q ( curl v 2 L 2 () + k4 ρ 2 v 2 L 2 ()). L k v, v H(curl,) curl v 2 L 2 () k2 v 2 L 2 () + 2 q curl v 2 L 2 () + 2k4 ρ 2 v 2 L q 2 () ) ( (1 + 2q curl v 2L2() k2 1 2k2 ρ 2 ) v 2 L q 2 (). Now let µ be an eigenvalue with corresponding eigenfunction ˆv V, ˆv, of the eigenvalue problem ] curl v curl ψ µvψ dx for all ψ V. (4.33) It is easily seen that such eigenvalues µ j exist, that is not an eigenvalue and that they are real and positive and converge to infinity. Taking ψ ˆv we conclude that curl ˆv 2 dx µ ˆv 2 dx. Substituting v ˆv in the previous estimate we conclude that ) L kˆv, ˆv H(curl,) (1 + 2q µ k 2 + 2k4 ρ 2 ] ˆv 2 L q 2 (). (4.34) The bracket ] has exactly the form of the bracket in estimate (3.22) of the previous section. Therefore, under the condition (3.23) and the choice we have that L kˆv, ˆv H(curl,). k 2 q 4ρ 2 We summarize the result in the following theorem. Theorem 4.6. Let ρ > be a constant such that an estimate of Poincare s type holds for the space W, i.e. ψ L 2 () ρ curl ψ L 2 () for all ψ W. Furthermore, let µ > be some eigenvalue of the eigenvalue problem (4.33). Assume that q satisfies (3.23). Then there exists at least one transmission eigenvalue k 2. Again, the same perturbation arguments as for Corollary 3.6 show existence of at least m eigenvalues: Corollary 4.7. Let µ j, j 1, 2, 3,..., be the eigenvalues of (4.33). We assume that they are ordered as µ 1 µ 2 and they appear according to their multiplicity. If q satisfies (q + 2)µ m q2 8ρ 2

18 18 ANREAS KIRSCH then there exist at least m transmission eigenvalues (counted according to their multiplicities). Remark 2. The classical form of the eigenvalue problem (4.33) is to find µ and v and ϕ such that curl 2 v µv ϕ in, divv in, ν v on, ν v on. REFERENCES 1] F. Cakoni and. Colton, On the mathematical basis of the linear sampling method, Georgian Math. J., 1 (23), ] F. Cakoni and. Colton, Qualitative Methods in Inverse Scattering Theory. An Introduction., Springer-Verlag, Berlin, 26. 3] F. Cakoni,. Colton, and H. Haddar, The linear sampling method for anisotropic media, J. Comp. Appl. Math., 146 (22), ]. Colton, H. Haddar, and P. Monk, The linear sampling method for solving the electromagnetic inverse scattering problem, SIAM J. Sci. Comput., 24 (22), ]. Colton, H. Haddar, and M. Piana, The linear sampling method in inverse electromagnetic scattering theory, Inverse Problems, 19 (23), S15 S137. 6]. Colton, A. Kirsch, and L. Päivärinta, Far field patterns for acoustic waves in an inhomogeneous medium, SIAM J. Math. Anal., 2 (1989), ]. Colton, L. Päivärinta, and J. Sylvester, The interior transmission problem, Inverse Problems and Imaging, 1 (27), ] B. ietz, J.-P. Eckmann, C.-A. Pillet, U. Smilansky, and I. Ussishkin, Inside-outside duality for planar billards: A numerical study, Physical Review E, (1995), ] H. Haddar, The interior transmission problem for anisotropic Maxwell s equations and its applications to the inverse problem, Math. Meth. Appl. Sci., 27 (24), ] H. Haddar and P. Monk, The linear sampling method for solving the electromagnetic inverse medium problem, Inverse Problems, 18 (22), ] T. Kato, Perturbation theory for linear operators, Springer-Verlag, ] A. Kirsch, Factorization of the far field operator for the inhomogeneous medium case and an application in inverse scattering theory, Inverse Problems, 15 (1999), ] A. Kirsch, An integral equation approach and the interior transmission problem for Maxwell s equations, Inverse Problems and Imaging, 1 (27), ] A. Kirsch, An integral equation for the scattering problem for an anisotropic medium and the Factorization method, In Proceedings of the 8th int. workshop on mathematical methods in scattering theory and biomedical engineering, ] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, ] P. Monk, Finite Element Methods for Maxwell s Equations, Oxford University Press, ] L. Päivärinta and J. Sylvester, Transmission eigenvalues, SIAM J. Math. Anal., 4 (28), pp ] B. Simon, Quantum mechanics for Hamiltonians defined as quadratic forms, Princeton University Press, address: kirsch@math.uni-karlsruhe.de

The Interior Transmission Eigenvalue Problem for Maxwell s Equations

The Interior Transmission Eigenvalue Problem for Maxwell s Equations The Interior Transmission Eigenvalue Problem for Maxwell s Equations Andreas Kirsch MSRI 2010 epartment of Mathematics KIT University of the State of Baden-Württemberg and National Large-scale Research

More information

NEW RESULTS ON TRANSMISSION EIGENVALUES. Fioralba Cakoni. Drossos Gintides

NEW RESULTS ON TRANSMISSION EIGENVALUES. Fioralba Cakoni. Drossos Gintides Inverse Problems and Imaging Volume 0, No. 0, 0, 0 Web site: http://www.aimsciences.org NEW RESULTS ON TRANSMISSION EIGENVALUES Fioralba Cakoni epartment of Mathematical Sciences University of elaware

More information

Discreteness of Transmission Eigenvalues via Upper Triangular Compact Operators

Discreteness 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 information

The Factorization Method for Maxwell s Equations

The Factorization Method for Maxwell s Equations The Factorization Method for Maxwell s Equations Andreas Kirsch University of Karlsruhe Department of Mathematics 76128 Karlsruhe Germany December 15, 2004 Abstract: The factorization method can be applied

More information

Inverse Scattering Theory: Transmission Eigenvalues and Non-destructive Testing

Inverse Scattering Theory: Transmission Eigenvalues and Non-destructive Testing Inverse Scattering Theory: Transmission Eigenvalues and Non-destructive Testing Isaac Harris Texas A & M University College Station, Texas 77843-3368 iharris@math.tamu.edu Joint work with: F. Cakoni, H.

More information

THE INTERIOR TRANSMISSION PROBLEM FOR REGIONS WITH CAVITIES

THE INTERIOR TRANSMISSION PROBLEM FOR REGIONS WITH CAVITIES THE INTERIOR TRANSMISSION PROBLEM FOR REGIONS WITH CAVITIES FIORALBA CAKONI, AVI COLTON, AN HOUSSEM HAAR Abstract. We consider the interior transmission problem in the case when the inhomogeneous medium

More information

A coupled BEM and FEM for the interior transmission problem

A coupled BEM and FEM for the interior transmission problem A coupled BEM and FEM for the interior transmission problem George C. Hsiao, Liwei Xu, Fengshan Liu, Jiguang Sun Abstract The interior transmission problem (ITP) is a boundary value problem arising in

More information

The Inside-Outside Duality for Scattering Problems by Inhomogeneous Media

The Inside-Outside Duality for Scattering Problems by Inhomogeneous Media The Inside-Outside uality for Scattering Problems by Inhomogeneous Media Andreas Kirsch epartment of Mathematics Karlsruhe Institute of Technology (KIT) 76131 Karlsruhe Germany and Armin Lechleiter Center

More information

An eigenvalue method using multiple frequency data for inverse scattering problems

An eigenvalue method using multiple frequency data for inverse scattering problems An eigenvalue method using multiple frequency data for inverse scattering problems Jiguang Sun Abstract Dirichlet and transmission eigenvalues have important applications in qualitative methods in inverse

More information

Transmission Eigenvalues in Inverse Scattering Theory

Transmission Eigenvalues in Inverse Scattering Theory Transmission Eigenvalues in Inverse Scattering Theory Fioralba Cakoni Department of Mathematical Sciences University of Delaware Newark, DE 19716, USA email: cakoni@math.udel.edu Jointly with D. Colton,

More information

The Imaging of Anisotropic Media in Inverse Electromagnetic Scattering

The Imaging of Anisotropic Media in Inverse Electromagnetic Scattering The Imaging of Anisotropic Media in Inverse Electromagnetic Scattering Fioralba Cakoni Department of Mathematical Sciences University of Delaware Newark, DE 19716, USA email: cakoni@math.udel.edu Research

More information

Factorization method in inverse

Factorization method in inverse Title: Name: Affil./Addr.: Factorization method in inverse scattering Armin Lechleiter University of Bremen Zentrum für Technomathematik Bibliothekstr. 1 28359 Bremen Germany Phone: +49 (421) 218-63891

More information

Transmission Eigenvalues in Inverse Scattering Theory

Transmission Eigenvalues in Inverse Scattering Theory Transmission Eigenvalues in Inverse Scattering Theory David Colton Department of Mathematical Sciences University of Delaware Newark, DE 19716, USA email: colton@math.udel.edu Research supported by a grant

More information

The Factorization Method for a Class of Inverse Elliptic Problems

The Factorization Method for a Class of Inverse Elliptic Problems 1 The Factorization Method for a Class of Inverse Elliptic Problems Andreas Kirsch Mathematisches Institut II Universität Karlsruhe (TH), Germany email: kirsch@math.uni-karlsruhe.de Version of June 20,

More information

Estimation of transmission eigenvalues and the index of refraction from Cauchy data

Estimation of transmission eigenvalues and the index of refraction from Cauchy data Estimation of transmission eigenvalues and the index of refraction from Cauchy data Jiguang Sun Abstract Recently the transmission eigenvalue problem has come to play an important role and received a lot

More information

and finally, any second order divergence form elliptic operator

and finally, any second order divergence form elliptic operator Supporting Information: Mathematical proofs Preliminaries Let be an arbitrary bounded open set in R n and let L be any elliptic differential operator associated to a symmetric positive bilinear form B

More information

The Factorization Method for Inverse Scattering Problems Part I

The 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 Baden-Württemberg and National Large-scale Research Center

More information

Transmission eigenvalues and non-destructive testing of anisotropic magnetic materials with voids

Transmission eigenvalues and non-destructive testing of anisotropic magnetic materials with voids Transmission eigenvalues and non-destructive testing of anisotropic magnetic materials with voids Isaac Harris 1, Fioralba Cakoni 1 and Jiguang Sun 2 1 epartment of Mathematical Sciences, University of

More information

On Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma. Ben Schweizer 1

On Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma. Ben Schweizer 1 On Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma Ben Schweizer 1 January 16, 2017 Abstract: We study connections between four different types of results that

More information

STOKES PROBLEM WITH SEVERAL TYPES OF BOUNDARY CONDITIONS IN AN EXTERIOR DOMAIN

STOKES PROBLEM WITH SEVERAL TYPES OF BOUNDARY CONDITIONS IN AN EXTERIOR DOMAIN Electronic Journal of Differential Equations, Vol. 2013 2013, No. 196, pp. 1 28. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu STOKES PROBLEM

More information

LECTURE 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 LECTURE 5 APPLICATIONS OF BDIE METHOD: ACOUSTIC SCATTERING BY INHOMOGENEOUS ANISOTROPIC OBSTACLES DAVID NATROSHVILI Georgian Technical University Tbilisi, GEORGIA 0-0 1. Formulation of the corresponding

More information

This article was published in an Elsevier journal. The attached copy is furnished to the author for non-commercial research and education use, including for instruction at the author s institution, sharing

More information

SPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS

SPECTRAL 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 information

Traces and Duality Lemma

Traces and Duality Lemma Traces and Duality Lemma Recall the duality lemma with H / ( ) := γ 0 (H ()) defined as the trace space of H () endowed with minimal extension norm; i.e., for w H / ( ) L ( ), w H / ( ) = min{ ŵ H () ŵ

More information

Notes on Transmission Eigenvalues

Notes on Transmission Eigenvalues Notes on Transmission Eigenvalues Cédric Bellis December 28, 2011 Contents 1 Scattering by inhomogeneous medium 1 2 Inverse scattering via the linear sampling method 2 2.1 Relationship with the solution

More information

ON THE MATHEMATICAL BASIS OF THE LINEAR SAMPLING METHOD

ON THE MATHEMATICAL BASIS OF THE LINEAR SAMPLING METHOD Georgian Mathematical Journal Volume 10 (2003), Number 3, 411 425 ON THE MATHEMATICAL BASIS OF THE LINEAR SAMPLING METHOD FIORALBA CAKONI AND DAVID COLTON Dedicated to the memory of Professor Victor Kupradze

More information

SELF-ADJOINTNESS OF SCHRÖDINGER-TYPE OPERATORS WITH SINGULAR POTENTIALS ON MANIFOLDS OF BOUNDED GEOMETRY

SELF-ADJOINTNESS OF SCHRÖDINGER-TYPE OPERATORS WITH SINGULAR POTENTIALS ON MANIFOLDS OF BOUNDED GEOMETRY Electronic Journal of Differential Equations, Vol. 2003(2003), No.??, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) SELF-ADJOINTNESS

More information

THE L 2 -HODGE THEORY AND REPRESENTATION ON R n

THE L 2 -HODGE THEORY AND REPRESENTATION ON R n THE L 2 -HODGE THEORY AND REPRESENTATION ON R n BAISHENG YAN Abstract. We present an elementary L 2 -Hodge theory on whole R n based on the minimization principle of the calculus of variations and some

More information

A 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 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 information

SPECTRAL APPROXIMATION TO A TRANSMISSION EIGENVALUE PROBLEM AND ITS APPLICATIONS TO AN INVERSE PROBLEM

SPECTRAL APPROXIMATION TO A TRANSMISSION EIGENVALUE PROBLEM AND ITS APPLICATIONS TO AN INVERSE PROBLEM SPECTRAL APPROXIMATIO TO A TRASMISSIO EIGEVALUE PROBLEM AD ITS APPLICATIOS TO A IVERSE PROBLEM JIG A JIE SHE 2,3 Abstract. We first develop an efficient spectral-galerkin method and an rigorous error analysis

More information

On the Spectrum of Volume Integral Operators in Acoustic Scattering

On the Spectrum of Volume Integral Operators in Acoustic Scattering 11 On the Spectrum of Volume Integral Operators in Acoustic Scattering M. Costabel IRMAR, Université de Rennes 1, France; martin.costabel@univ-rennes1.fr 11.1 Volume Integral Equations in Acoustic Scattering

More information

Homogenization of the Transmission Eigenvalue Problem for a Periodic Media

Homogenization of the Transmission Eigenvalue Problem for a Periodic Media Homogenization of the Transmission Eigenvalue Problem for a Periodic Media Isaac Harris Texas A&M University, Department of Mathematics College Station, Texas 77843-3368 iharris@math.tamu.edu Joint work

More information

Transmission eigenvalues with artificial background for explicit material index identification

Transmission eigenvalues with artificial background for explicit material index identification Transmission eigenvalues with artificial background for explicit material index identification Lorenzo Audibert 1,, Lucas Chesnel, Houssem Haddar 1 Department STEP, EDF R&D, 6 quai Watier, 78401, Chatou

More information

Lecture Note III: Least-Squares Method

Lecture Note III: Least-Squares Method Lecture Note III: Least-Squares Method Zhiqiang Cai October 4, 004 In this chapter, we shall present least-squares methods for second-order scalar partial differential equations, elastic equations of solids,

More information

SHARP BOUNDARY TRACE INEQUALITIES. 1. Introduction

SHARP 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 information

Title: Localized self-adjointness of Schrödinger-type operators on Riemannian manifolds. Proposed running head: Schrödinger-type operators on

Title: Localized self-adjointness of Schrödinger-type operators on Riemannian manifolds. Proposed running head: Schrödinger-type operators on Title: Localized self-adjointness of Schrödinger-type operators on Riemannian manifolds. Proposed running head: Schrödinger-type operators on manifolds. Author: Ognjen Milatovic Department Address: Department

More information

A modification of the factorization method for scatterers with different physical properties

A modification of the factorization method for scatterers with different physical properties A modification of the factorization method for scatterers with different physical properties Takashi FURUYA arxiv:1802.05404v2 [math.ap] 25 Oct 2018 Abstract We study an inverse acoustic scattering problem

More information

Numerical Solutions to Partial Differential Equations

Numerical 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 information

Finite Element Methods for Maxwell Equations

Finite Element Methods for Maxwell Equations CHAPTER 8 Finite Element Methods for Maxwell Equations The Maxwell equations comprise four first-order partial differential equations linking the fundamental electromagnetic quantities, the electric field

More information

SPECTRAL REPRESENTATIONS, AND APPROXIMATIONS, OF DIVERGENCE-FREE VECTOR FIELDS. 1. Introduction

SPECTRAL REPRESENTATIONS, AND APPROXIMATIONS, OF DIVERGENCE-FREE VECTOR FIELDS. 1. Introduction SPECTRAL REPRESENTATIONS, AND APPROXIMATIONS, OF DIVERGENCE-FREE VECTOR FIELDS GILES AUCHMUTY AND DOUGLAS R. SIMPKINS Abstract. Special solutions of the equation for a solenoidal vector field subject to

More information

The Helmholtz Equation

The Helmholtz Equation The Helmholtz Equation Seminar BEM on Wave Scattering Rene Rühr ETH Zürich October 28, 2010 Outline Steklov-Poincare Operator Helmholtz Equation: From the Wave equation to Radiation condition Uniqueness

More information

ELECTROMAGNETIC SCATTERING FROM PERTURBED SURFACES. Katharine Ott Advisor: Irina Mitrea Department of Mathematics University of Virginia

ELECTROMAGNETIC SCATTERING FROM PERTURBED SURFACES. Katharine Ott Advisor: Irina Mitrea Department of Mathematics University of Virginia ELECTROMAGNETIC SCATTERING FROM PERTURBED SURFACES Katharine Ott Advisor: Irina Mitrea Department of Mathematics University of Virginia Abstract This paper is concerned with the study of scattering of

More information

Complex geometrical optics solutions for Lipschitz conductivities

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 information

u xx + u yy = 0. (5.1)

u xx + u yy = 0. (5.1) Chapter 5 Laplace Equation The following equation is called Laplace equation in two independent variables x, y: The non-homogeneous problem u xx + u yy =. (5.1) u xx + u yy = F, (5.) where F is a function

More information

Second Order Elliptic PDE

Second 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 information

Variational Formulations

Variational 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 information

LECTURE 1: SOURCES OF ERRORS MATHEMATICAL TOOLS A PRIORI ERROR ESTIMATES. Sergey Korotov,

LECTURE 1: SOURCES OF ERRORS MATHEMATICAL TOOLS A PRIORI ERROR ESTIMATES. Sergey Korotov, LECTURE 1: SOURCES OF ERRORS MATHEMATICAL TOOLS A PRIORI ERROR ESTIMATES Sergey Korotov, Institute of Mathematics Helsinki University of Technology, Finland Academy of Finland 1 Main Problem in Mathematical

More information

THE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS

THE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS THE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS OGNJEN MILATOVIC Abstract. We consider H V = M +V, where (M, g) is a Riemannian manifold (not necessarily

More information

IMPROVED LEAST-SQUARES ERROR ESTIMATES FOR SCALAR HYPERBOLIC PROBLEMS, 1

IMPROVED LEAST-SQUARES ERROR ESTIMATES FOR SCALAR HYPERBOLIC PROBLEMS, 1 Computational Methods in Applied Mathematics Vol. 1, No. 1(2001) 1 8 c Institute of Mathematics IMPROVED LEAST-SQUARES ERROR ESTIMATES FOR SCALAR HYPERBOLIC PROBLEMS, 1 P.B. BOCHEV E-mail: bochev@uta.edu

More information

Weak Formulation of Elliptic BVP s

Weak Formulation of Elliptic BVP s Weak Formulation of Elliptic BVP s There are a large number of problems of physical interest that can be formulated in the abstract setting in which the Lax-Milgram lemma is applied to an equation expressed

More information

ON LIQUID CRYSTAL FLOWS WITH FREE-SLIP BOUNDARY CONDITIONS. Chun Liu and Jie Shen

ON LIQUID CRYSTAL FLOWS WITH FREE-SLIP BOUNDARY CONDITIONS. Chun Liu and Jie Shen DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Volume 7, Number2, April2001 pp. 307 318 ON LIQUID CRYSTAL FLOWS WITH FREE-SLIP BOUNDARY CONDITIONS Chun Liu and Jie Shen Department

More information

ITERATIVE METHODS FOR TRANSMISSION EIGENVALUES

ITERATIVE METHODS FOR TRANSMISSION EIGENVALUES ITERATIVE METHODS FOR TRANSMISSION EIGENVALUES JIGUANG SUN Abstract. Transmission eigenvalues have important applications in inverse scattering theory. They can be used to obtain useful information of

More information

Convex Hodge Decomposition of Image Flows

Convex Hodge Decomposition of Image Flows Convex Hodge Decomposition of Image Flows Jing Yuan 1, Gabriele Steidl 2, Christoph Schnörr 1 1 Image and Pattern Analysis Group, Heidelberg Collaboratory for Image Processing, University of Heidelberg,

More information

REMARKS ON THE VANISHING OBSTACLE LIMIT FOR A 3D VISCOUS INCOMPRESSIBLE FLUID

REMARKS ON THE VANISHING OBSTACLE LIMIT FOR A 3D VISCOUS INCOMPRESSIBLE FLUID REMARKS ON THE VANISHING OBSTACLE LIMIT FOR A 3D VISCOUS INCOMPRESSIBLE FLUID DRAGOŞ IFTIMIE AND JAMES P. KELLIHER Abstract. In [Math. Ann. 336 (2006), 449-489] the authors consider the two dimensional

More information

Computation of Maxwell s transmission eigenvalues and its applications in inverse medium problems

Computation of Maxwell s transmission eigenvalues and its applications in inverse medium problems Computation of Maxwell s transmission eigenvalues and its applications in inverse medium problems Jiguang Sun 1 and Liwei Xu 2,3 1 Department of Mathematical Sciences, Michigan Technological University,

More information

MULTIGRID PRECONDITIONING IN H(div) ON NON-CONVEX POLYGONS* Dedicated to Professor Jim Douglas, Jr. on the occasion of his seventieth birthday.

MULTIGRID PRECONDITIONING IN H(div) ON NON-CONVEX POLYGONS* Dedicated to Professor Jim Douglas, Jr. on the occasion of his seventieth birthday. MULTIGRID PRECONDITIONING IN H(div) ON NON-CONVEX POLYGONS* DOUGLAS N ARNOLD, RICHARD S FALK, and RAGNAR WINTHER Dedicated to Professor Jim Douglas, Jr on the occasion of his seventieth birthday Abstract

More information

Inequalities of Babuška-Aziz and Friedrichs-Velte for differential forms

Inequalities of Babuška-Aziz and Friedrichs-Velte for differential forms Inequalities of Babuška-Aziz and Friedrichs-Velte for differential forms Martin Costabel Abstract. For sufficiently smooth bounded plane domains, the equivalence between the inequalities of Babuška Aziz

More information

Numerical Analysis of Nonlinear Multiharmonic Eddy Current Problems

Numerical Analysis of Nonlinear Multiharmonic Eddy Current Problems Numerical Analysis of Nonlinear Multiharmonic Eddy Current Problems F. Bachinger U. Langer J. Schöberl April 2004 Abstract This work provides a complete analysis of eddy current problems, ranging from

More information

NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian UNDER NONHOMOGENEOUS NEUMANN BOUNDARY CONDITION

NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian UNDER NONHOMOGENEOUS NEUMANN BOUNDARY CONDITION Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 210, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian

More information

Uniqueness in determining refractive indices by formally determined far-field data

Uniqueness in determining refractive indices by formally determined far-field 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 far-field data Guanghui Hu a, Jingzhi

More information

An ill-posed boundary value problem for the Helmholtz equation on Lipschitz domains

An ill-posed boundary value problem for the Helmholtz equation on Lipschitz domains c de Gruyter 2009 J. Inv. Ill-Posed Problems 17 (2009), 703 711 DOI 10.1515 / JIIP.2009.041 An ill-posed boundary value problem for the Helmholtz equation on Lipschitz domains W. Arendt and T. Regińska

More information

LORENTZ SPACE ESTIMATES FOR VECTOR FIELDS WITH DIVERGENCE AND CURL IN HARDY SPACES

LORENTZ SPACE ESTIMATES FOR VECTOR FIELDS WITH DIVERGENCE AND CURL IN HARDY SPACES - TAMKANG JOURNAL OF MATHEMATICS Volume 47, Number 2, 249-260, June 2016 doi:10.5556/j.tkjm.47.2016.1932 This paper is available online at http://journals.math.tku.edu.tw/index.php/tkjm/pages/view/onlinefirst

More information

Mixed exterior Laplace s problem

Mixed exterior Laplace s problem Mixed exterior Laplace s problem Chérif Amrouche, Florian Bonzom Laboratoire de mathématiques appliquées, CNRS UMR 5142, Université de Pau et des Pays de l Adour, IPRA, Avenue de l Université, 64000 Pau

More information

Regularity of Weak Solution to an p curl-system

Regularity of Weak Solution to an p curl-system !#"%$ & ' ")( * +!-,#. /10 24353768:9 ;=A@CBEDGFIHKJML NPO Q

More information

L<MON P QSRTP U V!WYX7ZP U

L<MON P QSRTP U V!WYX7ZP U ! "$# %'&'(*) +,+.-*)%0/21 3 %4/5)6#7#78 9*+287:;)

More information

On uniqueness in the inverse conductivity problem with local data

On uniqueness in the inverse conductivity problem with local data On uniqueness in the inverse conductivity problem with local data Victor Isakov June 21, 2006 1 Introduction The inverse condictivity problem with many boundary measurements consists of recovery of conductivity

More information

EXISTENCE AND REGULARITY OF SOLUTIONS FOR STOKES SYSTEMS WITH NON-SMOOTH BOUNDARY DATA IN A POLYHEDRON

EXISTENCE AND REGULARITY OF SOLUTIONS FOR STOKES SYSTEMS WITH NON-SMOOTH BOUNDARY DATA IN A POLYHEDRON Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 147, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE AND REGULARITY OF SOLUTIONS FOR

More information

A new method for the solution of scattering problems

A new method for the solution of scattering problems A new method for the solution of scattering problems Thorsten Hohage, Frank Schmidt and Lin Zschiedrich Konrad-Zuse-Zentrum Berlin, hohage@zibde * after February 22: University of Göttingen Abstract We

More information

ON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM

ON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM ON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM OLEG ZUBELEVICH DEPARTMENT OF MATHEMATICS THE BUDGET AND TREASURY ACADEMY OF THE MINISTRY OF FINANCE OF THE RUSSIAN FEDERATION 7, ZLATOUSTINSKY MALIY PER.,

More information

Functional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability...

Functional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability... Functional Analysis Franck Sueur 2018-2019 Contents 1 Metric spaces 1 1.1 Definitions........................................ 1 1.2 Completeness...................................... 3 1.3 Compactness......................................

More information

Chapter 2 Finite Element Spaces for Linear Saddle Point Problems

Chapter 2 Finite Element Spaces for Linear Saddle Point Problems Chapter 2 Finite Element Spaces for Linear Saddle Point Problems Remark 2.1. Motivation. This chapter deals with the first difficulty inherent to the incompressible Navier Stokes equations, see Remark

More information

Mathematical Foundations for the Boundary- Field Equation Methods in Acoustic and Electromagnetic Scattering

Mathematical Foundations for the Boundary- Field Equation Methods in Acoustic and Electromagnetic Scattering Mathematical Foundations for the Boundary- Field Equation Methods in Acoustic and Electromagnetic Scattering George C. Hsiao Abstract The essence of the boundary-field equation method is the reduction

More information

FINITE ENERGY SOLUTIONS OF MIXED 3D DIV-CURL SYSTEMS

FINITE ENERGY SOLUTIONS OF MIXED 3D DIV-CURL SYSTEMS FINITE ENERGY SOLUTIONS OF MIXED 3D DIV-CURL SYSTEMS GILES AUCHMUTY AND JAMES C. ALEXANDER Abstract. This paper describes the existence and representation of certain finite energy (L 2 -) solutions of

More information

A Direct Method for reconstructing inclusions from Electrostatic Data

A Direct Method for reconstructing inclusions from Electrostatic Data A Direct Method for reconstructing inclusions from Electrostatic Data Isaac Harris Texas A&M University, Department of Mathematics College Station, Texas 77843-3368 iharris@math.tamu.edu Joint work with:

More information

Monotonicity-based inverse scattering

Monotonicity-based inverse scattering Monotonicity-based inverse scattering Bastian von Harrach http://numerical.solutions Institute of Mathematics, Goethe University Frankfurt, Germany (joint work with M. Salo and V. Pohjola, University of

More information

ENERGY NORM A POSTERIORI ERROR ESTIMATES FOR MIXED FINITE ELEMENT METHODS

ENERGY NORM A POSTERIORI ERROR ESTIMATES FOR MIXED FINITE ELEMENT METHODS ENERGY NORM A POSTERIORI ERROR ESTIMATES FOR MIXED FINITE ELEMENT METHODS CARLO LOVADINA AND ROLF STENBERG Abstract The paper deals with the a-posteriori error analysis of mixed finite element methods

More information

Volume and surface integral equations for electromagnetic scattering by a dielectric body

Volume and surface integral equations for electromagnetic scattering by a dielectric body Volume and surface integral equations for electromagnetic scattering by a dielectric body M. Costabel, E. Darrigrand, and E. H. Koné IRMAR, Université de Rennes 1,Campus de Beaulieu, 35042 Rennes, FRANCE

More information

The Asymptotic of Transmission Eigenvalues for a Domain with a Thin Coating

The Asymptotic of Transmission Eigenvalues for a Domain with a Thin Coating The Asymptotic of Transmission Eigenvalues for a Domain with a Thin Coating Hanen Boujlida, Houssem Haddar, Moez Khenissi To cite this version: Hanen Boujlida, Houssem Haddar, Moez Khenissi. The Asymptotic

More information

Stability of an abstract wave equation with delay and a Kelvin Voigt damping

Stability of an abstract wave equation with delay and a Kelvin Voigt damping Stability of an abstract wave equation with delay and a Kelvin Voigt damping University of Monastir/UPSAY/LMV-UVSQ Joint work with Serge Nicaise and Cristina Pignotti Outline 1 Problem The idea Stability

More information

Inverse Scattering Theory and Transmission Eigenvalues

Inverse Scattering Theory and Transmission Eigenvalues Inverse Scattering Theory and Transmission Eigenvalues David Colton Department of Mathematical Sciences University of Delaware Newark, DE 19716, USA email: colton@udel.edu Research supported a grant from

More information

1. Subspaces A subset M of Hilbert space H is a subspace of it is closed under the operation of forming linear combinations;i.e.,

1. Subspaces A subset M of Hilbert space H is a subspace of it is closed under the operation of forming linear combinations;i.e., Abstract Hilbert Space Results We have learned a little about the Hilbert spaces L U and and we have at least defined H 1 U and the scale of Hilbert spaces H p U. Now we are going to develop additional

More information

Technische Universität Graz

Technische Universität Graz Technische Universität Graz Stability of the Laplace single layer boundary integral operator in Sobolev spaces O. Steinbach Berichte aus dem Institut für Numerische Mathematik Bericht 2016/2 Technische

More information

Math 342 Partial Differential Equations «Viktor Grigoryan

Math 342 Partial Differential Equations «Viktor Grigoryan Math 342 Partial ifferential Equations «Viktor Grigoryan 3 Green s first identity Having studied Laplace s equation in regions with simple geometry, we now start developing some tools, which will lead

More information

Inverse scattering problem with underdetermined data.

Inverse scattering problem with underdetermined data. Math. Methods in Natur. Phenom. (MMNP), 9, N5, (2014), 244-253. Inverse scattering problem with underdetermined data. A. G. Ramm Mathematics epartment, Kansas State University, Manhattan, KS 66506-2602,

More information

EXISTENCE OF SOLUTIONS FOR A RESONANT PROBLEM UNDER LANDESMAN-LAZER CONDITIONS

EXISTENCE OF SOLUTIONS FOR A RESONANT PROBLEM UNDER LANDESMAN-LAZER CONDITIONS Electronic Journal of Differential Equations, Vol. 2008(2008), No. 98, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) EXISTENCE

More information

Chapter 1 Mathematical Foundations

Chapter 1 Mathematical Foundations Computational Electromagnetics; Chapter 1 1 Chapter 1 Mathematical Foundations 1.1 Maxwell s Equations Electromagnetic phenomena can be described by the electric field E, the electric induction D, the

More information

A priori error analysis of the BEM with graded meshes for the electric eld integral equation on polyhedral surfaces

A priori error analysis of the BEM with graded meshes for the electric eld integral equation on polyhedral surfaces A priori error analysis of the BEM with graded meshes for the electric eld integral equation on polyhedral surfaces A. Bespalov S. Nicaise Abstract The Galerkin boundary element discretisations of the

More information

A NOTE ON THE LADYŽENSKAJA-BABUŠKA-BREZZI CONDITION

A NOTE ON THE LADYŽENSKAJA-BABUŠKA-BREZZI CONDITION A NOTE ON THE LADYŽENSKAJA-BABUŠKA-BREZZI CONDITION JOHNNY GUZMÁN, ABNER J. SALGADO, AND FRANCISCO-JAVIER SAYAS Abstract. The analysis of finite-element-like Galerkin discretization techniques for the

More information

Appendix A Functional Analysis

Appendix A Functional Analysis Appendix A Functional Analysis A.1 Metric Spaces, Banach Spaces, and Hilbert Spaces Definition A.1. Metric space. Let X be a set. A map d : X X R is called metric on X if for all x,y,z X it is i) d(x,y)

More information

Coercivity of high-frequency scattering problems

Coercivity of high-frequency scattering problems Coercivity of high-frequency scattering problems Valery Smyshlyaev Department of Mathematics, University College London Joint work with: Euan Spence (Bath), Ilia Kamotski (UCL); Comm Pure Appl Math 2015.

More information

A NONCONFORMING PENALTY METHOD FOR A TWO DIMENSIONAL CURL-CURL PROBLEM

A NONCONFORMING PENALTY METHOD FOR A TWO DIMENSIONAL CURL-CURL PROBLEM A NONCONFORMING PENALTY METHOD FOR A TWO DIMENSIONAL CURL-CURL PROBLEM SUSANNE C. BRENNER, FENGYAN LI, AND LI-YENG SUNG Abstract. A nonconforming penalty method for a two-dimensional curl-curl problem

More information

Phys.Let A. 360, N1, (2006),

Phys.Let A. 360, N1, (2006), Phys.Let A. 360, N1, (006), -5. 1 Completeness of the set of scattering amplitudes A.G. Ramm Mathematics epartment, Kansas State University, Manhattan, KS 66506-60, USA ramm@math.ksu.edu Abstract Let f

More information

ON THE SPECTRUM OF NARROW NEUMANN WAVEGUIDE WITH PERIODICALLY DISTRIBUTED δ TRAPS

ON THE SPECTRUM OF NARROW NEUMANN WAVEGUIDE WITH PERIODICALLY DISTRIBUTED δ TRAPS ON THE SPECTRUM OF NARROW NEUMANN WAVEGUIDE WITH PERIODICALLY DISTRIBUTED δ TRAPS PAVEL EXNER 1 AND ANDRII KHRABUSTOVSKYI 2 Abstract. We analyze a family of singular Schrödinger operators describing a

More information

Math Theory of Partial Differential Equations Lecture 3-2: Spectral properties of the Laplacian. Bessel functions.

Math Theory of Partial Differential Equations Lecture 3-2: Spectral properties of the Laplacian. Bessel functions. Math 412-501 Theory of Partial ifferential Equations Lecture 3-2: Spectral properties of the Laplacian. Bessel functions. Eigenvalue problem: 2 φ + λφ = 0 in, ( αφ + β φ ) n = 0, where α, β are piecewise

More information

Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005

Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005 Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005 SOME INVERSE SCATTERING PROBLEMS FOR TWO-DIMENSIONAL SCHRÖDINGER

More information

i=1 α i. Given an m-times continuously

i=1 α i. Given an m-times 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 m-times continuously differentiable

More information

The Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation:

The Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: Oct. 1 The Dirichlet s P rinciple In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: 1. Dirichlet s Principle. u = in, u = g on. ( 1 ) If we multiply

More information

SELF-ADJOINTNESS OF DIRAC OPERATORS VIA HARDY-DIRAC INEQUALITIES

SELF-ADJOINTNESS OF DIRAC OPERATORS VIA HARDY-DIRAC INEQUALITIES SELF-ADJOINTNESS OF DIRAC OPERATORS VIA HARDY-DIRAC INEQUALITIES MARIA J. ESTEBAN 1 AND MICHAEL LOSS Abstract. Distinguished selfadjoint extension of Dirac operators are constructed for a class of potentials

More information

arxiv: v1 [math.ap] 21 Dec 2018

arxiv: v1 [math.ap] 21 Dec 2018 Uniqueness to Inverse Acoustic and Electromagnetic Scattering From Locally Perturbed Rough Surfaces Yu Zhao, Guanghui Hu, Baoqiang Yan arxiv:1812.09009v1 [math.ap] 21 Dec 2018 Abstract In this paper, we

More information

Perturbation Theory for Self-Adjoint Operators in Krein spaces

Perturbation Theory for Self-Adjoint Operators in Krein spaces Perturbation Theory for Self-Adjoint Operators in Krein spaces Carsten Trunk Institut für Mathematik, Technische Universität Ilmenau, Postfach 10 05 65, 98684 Ilmenau, Germany E-mail: carsten.trunk@tu-ilmenau.de

More information