The Interior Transmission Eigenvalue Problem for Maxwell s Equations
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1 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 Center of the Helmholtz Association
2 The Interior Transmission Eigenvalue Problem for Maxwell s Equations 2/17 Outline of the talk The irect Scattering Problem - A Tutorial The Interior Transmission Eigenvalue Problem An Example iscreteness of the Spectrum Existence of Eigenvalues
3 The Interior Transmission Eigenvalue Problem for Maxwell s Equations 3/17 The irect Scattering Problem - A Tutorial Formulation of the Scattering Problem: Given: ε = ε 0 ε r (x), µ = µ 0 and σ = 0 (i.e. non-magnetic and non-conducting media) such that ε r 1 has compact support. Given: Incident waves E inc, H inc which satisfy Maxwell system in frequency domain: etermine: curl E inc iωµ 0 H inc = 0 in R 3, curl H inc + iωε 0 E inc = 0 in R 3. Total fields E and H such that curl E iωµ 0 H = 0 in R 3, curl H + iωε 0 ε r E = 0 in R 3. E s = E E inc, H s = H H inc satisfy Silver-Müller radiation condition H s (x) x x E s (x) = O(1/ x ) as x.
4 The Interior Transmission Eigenvalue Problem for Maxwell s Equations 4/17 Formulation of the Scattering Problem Assumption: ε r L (R 3 ) with ε r = 1 outside some bounded domain and ε r c 0 > 0 on. Elimination of E: curl [ ε 1 r curl H ] k 2 H = 0 in R 3 with wave number k = ω ε 0 µ 0. The solution has to be understood in the weak sense, i.e. H H loc (curl, R 3 ) and [ (ε 1 r curl H) curl ψ k 2 H ψ ] dx = 0 R 3 for all ψ H(curl, R 3 ) with compact support. Remember: ψ H(curl, Ω) ψ, curl ψ L 2 (Ω) 3, ψ H loc (curl, R 3 ) ψ B H(curl, B) for all balls B
5 Volume Potential Fundamental solution of scalar Helmholtz equation u + k 2 u = 0: Φ k (x, y) = exp(ik x y ) 4π x y, x y. Theorem: efine volume potential with density a L 2 () 3 by V (x) = curl Φ k (x, y) a(y) dy = x Φ k (x, y) a(y) dy, x R 3. Then V H loc (curl, R 3 ) is the only radiating solution of in the weak sense, i.e. curl curl V k 2 V = curl a R 3 [ curl V curl ψ k 2 V ψ ] dx = for all ψ H(curl, R 3 ) with compact support. a curl ψ dx The Interior Transmission Eigenvalue Problem for Maxwell s Equations 5/17
6 The Interior Transmission Eigenvalue Problem for Maxwell s Equations 6/17 Integro-ifferential Equation Recall: Rewrite as: curl [ ε 1 r curl H ] k 2 H = 0 curl curl H k 2 H = curl [ (1 ε 1 r ) curl H ] }{{} = q Subtraction of curl curl H inc k 2 H inc = 0 yields curl curl H s k 2 H s = curl [ q curl H ], thus H(x) H inc (x) = H s (x) = curl Φ k (x, y) q(y) curl H(y) dy This is an integro-differential equation for H in H(curl, ). efine T k : H(curl, ) H(curl, ) by T k H = H curl Φ k (, y) q(y) curl H(y) dy
7 Fredholm Property Recall: T k H = H curl Φ k (, y) q(y) curl H(y) dy Then T i : H(curl, ) H(curl, ) is isomorphism onto. Indeed, T i H = V is equivalent to H = V + H and H satisfies H = curl Φ i (, y) q(y) [ curl H(y) + curl V (y) ] dy, thus curl curl H + H = curl [ q(curl H + curl V ) ], and in variational form R 3 [ (ε 1 r curl H) curl ψ + H ψ ] dx = for all ψ H(curl, R 3 ). (q curl V ) curl ψ dx Also T k T i is compact in H(curl, ). The Interior Transmission Eigenvalue Problem for Maxwell s Equations 7/17
8 The Interior Transmission Eigenvalue Problem for Maxwell s Equations 8/17 The irect Scattering Problem Thus: T k H = H inc T i H + (T k T i )H = H inc Theorem: (a) Uniqueness existence, continuous dependence (b) Uniqueness, existence, continuous dependence if ε r C 1,α () Asymptotic behavior from H s = curl Φ k(, y) q(y) curl H(y) dy: H s (x) = exp(ik x ) 4π x uniformly with respect to ˆx = x/ x S 2. H (ˆx) + O(1/ x 2 ), x, Far field pattern H satisfies H (ˆx) ˆx = 0 for all ˆx S 2.
9 The Interior Transmission Eigenvalue Problem for Maxwell s Equations 9/17 The Far Field Operator Incident field: H inc (x) = S 2 g(ˆθ) e ik ˆθ x ds(ˆθ), x R 3, with g L 2 t (S2 ) = {g L 2 (S 2 ) 3 : g(ˆθ) ˆθ = 0 on S 2 }. Corresponding far field pattern H defines far field operator F : L 2 t (S2 ) L 2 t (S2 ), g H. F is compact and normal, scattering matrix S = I + ik F is unitary. 8π 2 Theorem: If F is one-to-one and q q 0 > 0 on then the Factorization Method is applicable, i.e. for any fixed p R 3 with p 0 and any z R 3 define φ z L 2 t (S2 ) by φ z (ˆx) = (ˆx p) e ik ˆx z, ˆx S 2. Then: z φ z R ( (F F) 1/4)
10 The Interior Transmission Eigenvalue Problem for Maxwell s Equations 10/17 Interior Transmission Eigenvalue Problem Question: When is F one-to-one? Fg = 0 = H = 0 = H s = 0 outside of = H = H inc outside of = Cauchy data of H and H inc coincide on. Thus H, H inc satisfy: curl [ ε 1 r curl H ] k 2 H = 0 in, curl curl H inc k 2 H inc = 0 in, ν H = ν H inc, ν [ ε 1 r curl H ] = ν curl H inc on. efinition: λ > 0 is interior transmission eigenvalue if there exists non-trivial solution (u, w) H(curl, ) H(curl, ) with curl[p curl u] λ u = 0 in, curl curl w λ w = 0 in, ν u = ν w, ν [p curl u] = ν curl w on.
11 An Example curl [ p curl u ] λ u = 0, curl curl w λ w = 0 in, ν u = ν w, ν [p curl u] = ν curl w on. Example: unit ball p > 0 constant, λ = k 2. Ansatz (ρ = 1/ p): w(x) = α curl [ j n (kr)y n (ˆx) x ], u(x) = β curl [ j n (ρkr)y n (ˆx) x ] Boundary values: ν w = α j n (k) Grad Y n, ν curl w = α [ k j n(k) + j n (k) ] ˆx Grad Y n. Boundary conditions imply: ( j n (k) j n (ρk) k j n(k) + j n (k) p [ (ρk) j n(ρk) + j n (ρk) ] ) ( α β ) = ( 0 0 Asymptotic form of determinant yields existence of inifinitly many real eigenvalues λ j = k 2 j, and they converge to infinity. The Interior Transmission Eigenvalue Problem for Maxwell s Equations 11/17 ).
12 Elimination of w curl [ p curl u ] λ u = 0, curl curl w λ w = 0 in, ν u = ν w, ν [p curl u] = ν curl w on. Set v = w u and q = p 1. Assumption: q q 0 > 0 on curl [ p curl v ] λ v = curl [ q curl w ] in, ν v = 0, ν [p curl v] = ν [q curl w] on. Multiplication with ϕ for any ϕ H 1 () and integration yields: curl [ p curl v ] ϕ dx λ v ϕ dx = curl [ q curl w ] ϕ dx. Thus: div v = 0 in, ν v = 0 on, i.e. v X 0 where X 0 = { v H 0 (curl, ) : v ϕ dx = 0 ϕ H 1 () }. Also: X = { w H(curl, ) : The Interior Transmission Eigenvalue Problem for Maxwell s Equations 12/17 w ϕ dx = 0 ϕ H 1 () }.
13 Elimination of w, cont. Given v X 0, find w X with curl [ q curl w ] = curl [ p curl v ] λ v in, ν [q curl w] = ν [p curl v] on. This defines A λ : X 0 X0 by [ ] A λ v, ψ = curl w curl ψ λ w ψ dx, v, ψ X0. Theorem: (a) Let (u, w) solve interior transmission eigenvalue problem for some λ > 0. Then A λ v = 0 for v = w u. (b) Let v X 0 satisfy A λ v = 0. efine w X according to above. Then there exists φ H 1 () such that (u, w + φ) solves interior transmission eigenvalue problem where u = w + φ v. The Interior Transmission Eigenvalue Problem for Maxwell s Equations 13/17
14 iscreteness of Spectrum [ ] A λ v, ψ = curl w curl ψ λ w ψ dx, v, ψ X0. [ ] (q curl w) curl ψ dx = (p curl v) curl ψ λv ψ dx for all ψ X. Theorem: A λ : X 0 X 0 has the form A λ = A 0 + λc 1 + λ 2 C 2 where A 0 is self-adjoint and coercive and C 1, C 2 are self-adjoint and compact. Furthermore, C 2 is non-negative. Idea of proof: v j w j, j = 1, 2 (recall: p = q + 1) [ A λ v 1, ψ = (p curl w1 ) curl ψ λ w ] 1 ψ dx (q curl w 1 ) curl ψ dx Set u = w v : A 0 v, v = = curl v 2 dx + curl v 2 dx + curl u curl v dx (q curl u) curl u dx curl v 2 L 2 () The Interior Transmission Eigenvalue Problem for Maxwell s Equations 14/17
15 The Interior Transmission Eigenvalue Problem for Maxwell s Equations 15/17 iscreteness of Spectrum, cont. Quadratic eigenvalue problem: Find λ > 0 and v 0 with A 0 v + λc 1 v + λ 2 C 2 v = 0! Rewrite as (set v 1 = A 1/2 0 v): v 1 + λ A 1/2 0 C 1 A 1/2 0 v 1 + λ }{{} = K 1 2 A 1/2 0 } C 2 A 1/2 0 v 1 {{} = 0. = K2 2 With v 2 = λ K 2 v 1 this becomes ( ) ( v1 K1 K = λ 2 K 2 0 v 2 ) ( v1 v 2 ) This is non-self adjoint eigenvalue problem for compact operator. Therefore, spectrum is discrete, eigenvalues (if they exist) tend to infinity.
16 The Interior Transmission Eigenvalue Problem for Maxwell s Equations 16/17 Existence of Eigenvalues Goal: Find λ > 0 such that A λ fails to be one-to-one, i.e. find λ > 0 such that µ = 0 is eigenvalue of A λ. Let µ min (λ) be the smallest eigenvalue of A λ. Then µ min (0) > 0 since A 0 is coercive. Idea (Päivärinta, Sylvester 2008): Find ˆλ > 0 with µ min (ˆλ) 0. Then there exists λ (0, ˆλ) with µ min (λ) = 0. Idea (Cakoni, Gintides 2010): Choose ball B and eigenvalue ˆλ > 0 with eigenfunctions û, ŵ H 1 (B) corresponding to the ball and constant p 0 = 1 + q 0. Set ˆv = ŵ û. Then ˆv X 0 (B) and ˆλˆv = 0. Extend ˆv by zero into, then ˆv X 0 () and Aˆλˆv, ˆv... ˆλˆv, ˆv = 0. Therefore, µ min (ˆλ) 0. Refined arguments: There exist infinitely many real eigenvalues λ j.
17 Remarks Some References: Colton, Kirsch, Päivärinta: Far field patterns for acoustic waves in an inhomogeneous medium, 1989 u + k 2 pu = 0, w + k 2 w = 0 in, u = w, u ν = w ν Päivärinta, Sylvester: Transmission eigenvalues, 2008 on. Cakoni, Gintides, Haddar: The existence of an infinite discrete set of transmission eigenvalues, 2010 Cakoni, Colton, Haddar: The interior transmission problem for regions with cavity, 2010 Cakoni, Kirsch: On the interior transmission eigenvalue problem, 2010 Open problems: u + k 2 pu = 0 in \ B, u = 0 on B Existence of complex eigenvalues? rop assumption p > 1? The Interior Transmission Eigenvalue Problem for Maxwell s Equations 17/17
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