ON 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 of Mathematics 76128 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 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

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 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 2 1 + q (1 + q ) 2 /2)ρ +µ 1 (1 + q /2) 2 ρ 2. 1 + 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 2 1 + q which can be rewritten as (note that µ 1 ρ 2 by, e.g., 17]) ( ) ] µ1 µ1 µ1 q > 2 1 + 1. (2.9) ρ 2 ρ ρ 2

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 2 1 + ρ ρ 2 1 then there exist at least m transmission eigenvalues (counted according to their multiplicities).

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 ().

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 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

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

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.

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 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 ().

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 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.

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 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 ()

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 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), 95 14. 2] 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), 285 299. 4]. Colton, H. Haddar, and P. Monk, The linear sampling method for solving the electromagnetic inverse scattering problem, SIAM J. Sci. Comput., 24 (22), 719 731. 5]. 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), 1472 1483. 7]. Colton, L. Päivärinta, and J. Sylvester, The interior transmission problem, Inverse Problems and Imaging, 1 (27), 13 28. 8] 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), 4222 4234. 9] H. Haddar, The interior transmission problem for anisotropic Maxwell s equations and its applications to the inverse problem, Math. Meth. Appl. Sci., 27 (24), 2111 2129. 1] H. Haddar and P. Monk, The linear sampling method for solving the electromagnetic inverse medium problem, Inverse Problems, 18 (22), 891 96. 11] T. Kato, Perturbation theory for linear operators, Springer-Verlag, 1976. 12] A. Kirsch, Factorization of the far field operator for the inhomogeneous medium case and an application in inverse scattering theory, Inverse Problems, 15 (1999), 413 429. 13] A. Kirsch, An integral equation approach and the interior transmission problem for Maxwell s equations, Inverse Problems and Imaging, 1 (27), 159 18. 14] 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, 27. 15] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, 28. 16] P. Monk, Finite Element Methods for Maxwell s Equations, Oxford University Press, 23. 17] L. Päivärinta and J. Sylvester, Transmission eigenvalues, SIAM J. Math. Anal., 4 (28), pp. 738 753. 18] B. Simon, Quantum mechanics for Hamiltonians defined as quadratic forms, Princeton University Press, 1971. E-mail address: kirsch@math.uni-karlsruhe.de