SLIDE 1 Stability for the electromagnetic scattering problem
Luca RONDI
Università di Trieste
Joint work with Hongyu LIU and Jingni XIAO (Hong Kong Baptist University)
Workshop Analysis and Numerics of Acoustic and Electromagnetic Problems Linz, 17 – 22 October 2016
SLIDE 2
The electromagnetic scattering problem
Existence and uniqueness
SLIDE 3 Time-harmonic EM scattering problem: setting
Basic notation: Σ ⊂ R3 scatterer Σ compact, G = R3\Σ connected (G exterior domain) k > 0 wavenumber; ǫ electric permittivity; µ magnetic permeability Basic assumptions:
Fixed R0 > 0 and 0 < λ0 < 1 < λ1, we assume
Σ ⊂ BR0 ǫ, µ ∈ L∞(G, M3×3
sym(R)) such that
λ0I3 ǫ(x), µ(x) λ1I3
for a.e. x ∈ G and
ǫ(x) = µ(x) = I3
if x > R0.
SLIDE 4 Time-harmonic EM scattering problem
Σ perfectly electric conducting scatterer
Incident time-harmonic EM wave: incident electric and magnetic fields
(Ei, Hi), e.g. normalised electromagnetic plane wave Ei(x) = i k∇ ∧
, Hi(x) = ∇ ∧ peikx·d, x ∈ R3. p ∈ R3, p = 0, polarisation vector; d ∈ S2 incident direction
Find the total electric and magnetic fields (E, H), or the scattered electric and magnetic fields (Es, Hs), solving
∇ ∧ E − ikµH = 0, ∇ ∧ H + ikǫE = 0
in G = R3\Σ
(E, H) = (Ei, Hi) + (Es, Hs)
in G
ν ∧ E = 0
limr→+∞ r
x ∧ Hs(x) + Es(x)
r = x. Remark: analogous for Σ perfectly magnetic conducting scatterer, with
boundary condition ν ∧ H = 0 on ∂G = ∂Σ
SLIDE 5
EM scattering problem: existence and uniqueness (1)
Main ingredients Rellich Compactness Property (RCP) D bounded domain has RCP if the immersion of H1(D) into L2(D) is compact Maxwell Compactness Property (MCP) D bounded domain has MCP if the immersions of H0(curl, D) ∩ H(div, D) into L2(D, C3) and of H(curl, D) ∩ H0(div, D) into L2(D, C3) are compact Unique Continuation Property (UCP)
The Maxwell system in a domain D
∇ ∧ E − ikµH = 0
and
∇ ∧ H + ikǫE = 0
in D satisfies the UCP if (E, H) ≡ 0 on D whenever (E, H) ≡ 0 on an open nonempty subset of D
SLIDE 6 EM scattering problem: existence and uniqueness (2)
Σ ⊂ BR0 perfectly electric conducting scatterer
Incident electric and magnetic fields (Ei, Hi), e.g. normalised electromagnetic plane wave
Existence and uniqueness (Picard, Weck, Witsch (2001))
Assume that for some R > R0, G ∩ BR satisfies RCP and MCP. Assume that the Maxwell system satisfies UCP. Then
∇ ∧ E − ikµH = 0, ∇ ∧ H + ikǫE = 0
in G = R3\Σ
(E, H) = (Ei, Hi) + (Es, Hs)
in G
ν ∧ E = 0
limr→+∞ r
x ∧ Hs(x) + Es(x)
r = x
admits a unique solution.
SLIDE 7
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 Ux s. t. Ux ∩ D has a finite number of connected components
if A is a connected component of Ux ∩ D such that x ∈ ∂A, then ∃ a bi-W1,∞ mapping between A and a Lipschitz domain Then D satisfies both the RCP and MCP.
SLIDE 8
Examples
SLIDE 9
Examples
SLIDE 10
On UCP
∇ ∧ E − ikµH = 0, ∇ ∧ H + ikǫE = 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 {Di} s. t. for some s < 2 and up to a set of finite Hs measure, any x ∈ σ ∩ D separates exactly two partitions
(ǫ, µ)|Di is the restriction of locally Lipschitz coefficients in D
Then we have the UCP in D.
SLIDE 11
The electromagnetic scattering problem
Stability
SLIDE 12 EM scattering problem: stability issue
Σn ⊂ BR0 perfectly electric conducting scatterer; Σn → Σ∞ kn wavenumber; kn → k∞ > 0 ǫn electric permittivity, µn magnetic permeability; ǫn → ǫ∞, µn → µ∞ (Ei
n, Hi n) incident fields; pn polarisation vector, dn ∈ S2 incident direction
(Ei
n, Hi n) → (Ei ∞, Hi ∞); pn → p∞, dn → d∞ ∈ S2
(En, Hn), (Es
n, Hs n) solution to
∇ ∧ En − iknµnHn = 0, ∇ ∧ Hn + iknǫnEn = 0
in Gn = R3\Σn
(En, Hn) = (Ei
n, Hi n) + (Es n, Hs n)
in Gn
ν ∧ En = 0
limr→+∞ r
x ∧ Hs n(x) + Es n(x)
r = x Main question
Does (En, Hn) → (E∞, H∞)?
SLIDE 13
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
SLIDE 14
Basic bricks: classes of admissible sets and coefficients
Classes of admissible sets B: class of compact sets K ⊂ BR0, where K is the union of a finite
number of Lipschitz hypersurfaces (with or without boundary) intersecting nontangentially
C: class of compact sets Σ ⊂ BR0 such that ∂Σ ∈ B and Σ satisfies, in
a uniform quantitative way, the sufficient condition for RCP and MCP
Cscat: class of scatterers Σ ∈ C, such that G = R3\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 = I3 outside BR0
Remark: the class N is compact with respect to the Lp convergence, for
any 1 p < +∞
SLIDE 15 Mosco convergence
Elliptic equations
Convergence of solutions to Neumann problems with respect to variations
⇒ Mosco convergence of corresponding H1 spaces
Mosco convergence of H1 spaces
∂Dn → ∂D in the Hausdorff distance
= ⇒
H1(Dn) → H1(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: ∂Dn union of a bounded number of Lipschitz
hypersurfaces either not-intersecting (Giacomini (2004)) or intersecting nontangentially, i.e. ∂Dn ∈ B, (Menegatti & Rondi (2013))
SLIDE 16
Acoustic scattering problems with sound-hard scatterers
Helmholtz equation (Menegatti & Rondi (2013))
Mosco convergence of H1 spaces & uniform higher integrability property for H1 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 H1 functions
For any Σ ∈ Cscat, for some s > 2,
vLs(BR0+1\Σ) C1vH1(BR0+1\Σ)
for any v ∈ H1(BR0+1\Σ)
SLIDE 17
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 ∂Dn ∈ B. Then
∂Dn → ∂D in the Hausdorff distance
= ⇒
H(curl, Dn) → H(curl, , D) in the Mosco sense
SLIDE 18 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 ∈ Cscat, if (E, H) solves
∇ ∧ E − ikµH = 0, ∇ ∧ H + ikǫE = 0
in BR0+1\Σ
ν ∧ E = 0
then, for some s > 2,
ELs(BR0+1\Σ)+HLs(BR0+1\Σ) C1
- EL2(BR0+1\Σ)+HL2(BR0+1\Σ)
+ ν ∧ EL2(∂BR0+1) + ν ∧ HL2(∂BR0+1)
- Remark: no regularity assumptions on the coefficients ǫ, µ
SLIDE 19 Stability result for EM scattering problems
The main stability result Σn ∈ Cscat; Σn → Σ∞ in the Hausdorff distance 0 < k kn k; kn → k∞ (ǫn, µn) ∈ N; (ǫn, µn) → (ǫ∞, µ∞) in Lp, for any 1 p < +∞ (Ei
n, Hi n) incident fields, with pn ∈ R3, pn 1, dn ∈ S2;
pn → p∞, dn → d∞ ∈ S2 (En, Hn), (Es
n, Hs n) solution to
∇ ∧ En − iknµnHn = 0, ∇ ∧ Hn + iknǫnEn = 0
in Gn = R3\Σn
(En, Hn) = (Ei
n, Hi n) + (Es n, Hs n)
in Gn
ν ∧ En = 0
limr→+∞ r
x ∧ Hs n(x) + Es n(x)
r = x
Then, locally in L2,
(En, Hn) → (E∞, H∞)
and
(∇ ∧ En, ∇ ∧ Hn) → (∇ ∧ E∞, ∇ ∧ H∞)
SLIDE 20 Uniform estimates w. r. t. scatterers and inhomogeneities
Uniform estimates for solutions to EM scattering problems Σ ∈ Cscat; 0 < k k k; (ǫ, µ) ∈ N (Ei, Hi) incident fields, with p ∈ R3, p 1, d ∈ S2 (E, H), (Es, Hs) solution to ∇ ∧ E − ikµH = 0, ∇ ∧ H + ikǫE = 0
in G = R3\Σ
(E, H) = (Ei, Hi) + (Es, Hs)
in G
ν ∧ E = 0
limr→+∞ r
x ∧ Hs(x) + Es(x)
r = x
Then, for any R > 0, ∃ constant C s.t.
EL2(BR\Σ) + HL2(BR\Σ) C.
Furthermore, ∃ constant C1 s.t.
Es(x) + Hs(x) C1x−1
if x R0 + 1/2.
