Stability for the electromagnetic scattering problem Luca R ONDI - PowerPoint PPT Presentation

Stability for the electromagnetic scattering problem Luca R ONDI Università di Trieste Joint work with Hongyu L IU and Jingni X IAO (Hong Kong Baptist University) Workshop Analysis and Numerics of Acoustic and Electromagnetic Problems Linz, 17 – 22 October 2016

The electromagnetic scattering problem Existence and uniqueness

Time-harmonic EM scattering problem: setting Basic notation: Σ ⊂ R 3 scatterer Σ compact, G = R 3 \ Σ connected ( G exterior domain) k > 0 wavenumber; ǫ electric permittivity; µ magnetic permeability Basic assumptions: Fixed R 0 > 0 and 0 < λ 0 < 1 < λ 1 , we assume Σ ⊂ B R 0 ǫ , µ ∈ L ∞ ( G , M 3 × 3 sym ( R )) such that λ 0 I 3 � ǫ ( x ) , µ ( x ) � λ 1 I 3 for a.e. x ∈ G and ǫ ( x ) = µ ( x ) = I 3 if � x � > R 0 .

Time-harmonic EM scattering problem Σ perfectly electric conducting scatterer Incident time-harmonic EM wave: incident electric and magnetic fields ( E i , H i ) , e.g. normalised electromagnetic plane wave E i ( x ) = i ∇ ∧ p e i k x · d � H i ( x ) = ∇ ∧ p e i k x · d , x ∈ R 3 . � k ∇ ∧ , p ∈ R 3 , p � = 0 , polarisation vector; d ∈ S 2 incident direction Find the total electric and magnetic fields ( E , H ) , or the scattered electric and magnetic fields ( E s , H s ) , solving  in G = R 3 \ Σ ∇ ∧ E − i kµ H = 0, ∇ ∧ H + i kǫ E = 0    ( E , H ) = ( E i , H i ) + ( E s , H s ) in G  ν ∧ E = 0 on ∂G = ∂Σ   � �  x � x � ∧ H s ( x ) + E s ( x )  lim r → + ∞ r = 0 r = � x � . Remark: analogous for Σ perfectly magnetic conducting scatterer, with boundary condition ν ∧ H = 0 on ∂G = ∂Σ

EM scattering problem: existence and uniqueness (1) Main ingredients Rellich Compactness Property (RCP) D bounded domain has RCP if the immersion of H 1 ( D ) into L 2 ( D ) is compact Maxwell Compactness Property (MCP) D bounded domain has MCP if the immersions of H 0 ( curl, D ) ∩ H ( div, D ) into L 2 ( D , C 3 ) and of H ( curl, D ) ∩ H 0 ( div, D ) into L 2 ( D , C 3 ) are compact Unique Continuation Property (UCP) The Maxwell system in a domain D ∇ ∧ E − i kµ H = 0 and ∇ ∧ H + i kǫ E = 0 in D satisfies the UCP if ( E , H ) ≡ 0 on D whenever ( E , H ) ≡ 0 on an open nonempty subset of D

EM scattering problem: existence and uniqueness (2) Σ ⊂ B R 0 perfectly electric conducting scatterer Incident electric and magnetic fields ( E i , H i ) , e.g. normalised electromagnetic plane wave Existence and uniqueness (Picard, Weck, Witsch (2001)) Assume that for some R > R 0 , G ∩ B R satisfies RCP and MCP. Assume that the Maxwell system satisfies UCP. Then  in G = R 3 \ Σ ∇ ∧ E − i kµ H = 0, ∇ ∧ H + i kǫ E = 0    ( E , H ) = ( E i , H i ) + ( E s , H s )  in G ν ∧ E = 0 on ∂G = ∂Σ   � �  � x � ∧ H s ( x ) + E s ( x ) x  r = � x � lim r → + ∞ r = 0 admits a unique solution.

On RCP and MCP Sufficient conditions for RCP: classical Sufficient conditions for MCP: Picard, Weck, Witsch (2001) New sufficient condition for RCP and MCP D bounded domain. For any x ∈ ∂D , ∃ an open neighbourhood U x s. t. U x ∩ D has a finite number of connected components if A is a connected component of U x ∩ D such that x ∈ ∂A , then ∃ a bi- W 1, ∞ mapping between A and a Lipschitz domain Then D satisfies both the RCP and MCP.

Examples

Examples

On UCP ∇ ∧ E − i kµ H = 0, ∇ ∧ H + i kǫ E = 0 in a domain D Basic brick for UCP (Nguyen, Wang (2012)) If ǫ and µ are locally Lipschitz, then we have the UCP in D . Sufficient condition for UCP: Ball, Capdeboscq, Tsering-Xiao (2012) for piecewise Lipschitz coefficients Generalised condition for piecewise Lipschitz coefficients Up to a closed set σ , the discontinuity set, with | σ | = 0 , D is partitioned into pairwise disjoint domains { D i } s. t. for some s < 2 and up to a set of finite H s measure, any x ∈ σ ∩ D separates exactly two partitions ( ǫ , µ ) | D i is the restriction of locally Lipschitz coefficients in D Then we have the UCP in D .

The electromagnetic scattering problem Stability

EM scattering problem: stability issue Σ n ⊂ B R 0 perfectly electric conducting scatterer; Σ n → Σ ∞ k n wavenumber; k n → k ∞ > 0 ǫ n electric permittivity, µ n magnetic permeability; ǫ n → ǫ ∞ , µ n → µ ∞ n ) incident fields; p n polarisation vector, d n ∈ S 2 incident direction ( E i n , H i ( E i n , H i n ) → ( E i ∞ , H i ∞ ) ; p n → p ∞ , d n → d ∞ ∈ S 2 ( E n , H n ) , ( E s n , H s n ) solution to  in G n = R 3 \ Σ n ∇ ∧ E n − i k n µ n H n = 0, ∇ ∧ H n + i k n ǫ n E n = 0    ( E n , H n ) = ( E i n , H i n ) + ( E s n , H s n ) in G n  ν ∧ E n = 0 on ∂G n = ∂Σ n   � �  x � x � ∧ H s n ( x ) + E s  lim r → + ∞ r n ( x ) = 0 r = � x � Main question Does ( E n , H n ) → ( E ∞ , H ∞ ) ?

Stability for EM scattering problem: main issues Find class of scatterers for which we have: compactness the boundary condition ν ∧ E = 0 is preserved at the limit ❀ Mosco convergence for H ( curl ) spaces quantitative versions of RCP and MCP that are uniform with respect to the scatterers ❀ higher integrability for solutions to Maxwell system in nonsmooth domains Find class of coefficients for which we have: compactness UCP holds for all coefficients

Basic bricks: classes of admissible sets and coefficients Classes of admissible sets B : class of compact sets K ⊂ B R 0 , where K is the union of a finite number of Lipschitz hypersurfaces (with or without boundary) intersecting nontangentially C : class of compact sets Σ ⊂ B R 0 such that ∂Σ ∈ B and Σ satisfies, in a uniform quantitative way, the sufficient condition for RCP and MCP C scat : class of scatterers Σ ∈ C , such that G = R 3 \ Σ satisfies, in a uniform quantitative way, a connectedness condition Remark: all classes are compact with respect to the Hausdorff distance Classes of coefficients N : class of piecewise Lipschitz coefficients ǫ and µ whose discontinuity set σ ∈ B , with ǫ = µ = I 3 outside B R 0 Remark: the class N is compact with respect to the L p convergence, for any 1 � p < + ∞

Mosco convergence Elliptic equations Convergence of solutions to Neumann problems with respect to variations ⇒ Mosco convergence of corresponding H 1 spaces of the domain ⇐ Mosco convergence of H 1 spaces ∂D n → ∂D in the Hausdorff distance H 1 ( D n ) → H 1 ( D ) in the Mosco sense ? ⇒ = N = 2 by duality arguments Chambolle & Doveri (1997): sufficient condition Bucur & Varchon (2000): necessary and sufficient condition N = 3 sufficient condition: ∂D n union of a bounded number of Lipschitz hypersurfaces either not-intersecting (Giacomini (2004)) or intersecting nontangentially, i.e. ∂D n ∈ B , (Menegatti & Rondi (2013))

Acoustic scattering problems with sound-hard scatterers Helmholtz equation (Menegatti & Rondi (2013)) Mosco convergence of H 1 spaces & uniform higher integrability property for H 1 functions ⇒ = convergence of solutions to Neumann problems for the Helmholtz equation with respect to variations of the domain convergence of solutions to acoustic scattering problems with respect to variations of the sound-hard scatterer Higher integrability in nonsmooth domains for H 1 functions For any Σ ∈ C scat , for some s > 2 , for any v ∈ H 1 ( B R 0 + 1 \ Σ ) � v � L s ( B R 0 + 1 \ Σ ) � C 1 � v � H 1 ( B R 0 + 1 \ Σ )

Electromagnetic scattering problems Maxwell system Mosco convergence of H ( curl ) spaces & uniform higher integrability property for solutions to Maxwell system ⇒ = convergence of solutions to EM scattering problems with respect to variations of the scatterer Mosco convergence for H ( curl ) spaces Let ∂D n ∈ B . Then ∂D n → ∂D in the Hausdorff distance ⇒ H ( curl, D n ) → H ( curl, , D ) in the Mosco sense =

Higher integrability for solutions to Maxwell system Basic brick for higher integrability for Maxwell Druet (2012) for Lipschitz domains Higher integrability in nonsmooth domains for Maxwell For any Σ ∈ C scat , if ( E , H ) solves � ∇ ∧ E − i kµ H = 0, ∇ ∧ H + i kǫ E = 0 in B R 0 + 1 \ Σ ν ∧ E = 0 on ∂Σ then, for some s > 2 , � � E � L s ( B R 0 + 1 \ Σ ) + � H � L s ( B R 0 + 1 \ Σ ) � C 1 � E � L 2 ( B R 0 + 1 \ Σ ) + � H � L 2 ( B R 0 + 1 \ Σ ) � + � ν ∧ E � L 2 ( ∂B R 0 + 1 ) + � ν ∧ H � L 2 ( ∂B R 0 + 1 ) Remark: no regularity assumptions on the coefficients ǫ , µ

Stability result for EM scattering problems The main stability result Σ n ∈ C scat ; Σ n → Σ ∞ in the Hausdorff distance 0 < k � k n � k ; k n → k ∞ ( ǫ n , µ n ) ∈ N ; ( ǫ n , µ n ) → ( ǫ ∞ , µ ∞ ) in L p , for any 1 � p < + ∞ ( E i n , H i n ) incident fields, with p n ∈ R 3 , � p n � � 1 , d n ∈ S 2 ; p n → p ∞ , d n → d ∞ ∈ S 2 ( E n , H n ) , ( E s n , H s n ) solution to  in G n = R 3 \ Σ n ∇ ∧ E n − i k n µ n H n = 0, ∇ ∧ H n + i k n ǫ n E n = 0    ( E n , H n ) = ( E i n , H i n ) + ( E s n , H s n ) in G n  ν ∧ E n = 0 on ∂G n = ∂Σ n   � �  � x � ∧ H s x n ( x ) + E s  n ( x ) = 0 r = � x � lim r → + ∞ r Then, locally in L 2 , ( E n , H n ) → ( E ∞ , H ∞ ) and ( ∇ ∧ E n , ∇ ∧ H n ) → ( ∇ ∧ E ∞ , ∇ ∧ H ∞ )