10. Time-harmonic electro-magnetic scattering in exterior weak Lipschitz domains with mixed boundary conditions
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Frank Osterbrink
Abstract
This paper treats the time-harmonic electro-magnetic scattering or radiation problem governed by Maxwell’s equations, i. e., − rot H + iωεE = F in Ω, E × ν = 0 on Γ1, rot E + iωμH = G in Ω, H × ν = 0 on Γ2, where ω ∈ ℂ \ (0) and Ω ⊂ ℝ3 is an exterior weak Lipschitz domain with boundary Γ divided into two disjoint parts Γ1 and Γ2. We will present a solution theory using the framework of polynomially weighted Sobolev spaces for the rotation and divergence. For the physically interesting case ω ∈ ℝ \ (0), we will show a Fredholm alternative type result to hold using the principle of limiting absorption introduced by Eidus in the 1960s. The necessary a priori estimate and polynomial decay of eigenfunctions for the Maxwell equations will be obtained by transferring well-known results for the Helmholtz equation using a suitable decomposition of the fields E and H. The crucial point for existence is a local version of Weck’s selection theorem, also called Maxwell compactness property.
Abstract
This paper treats the time-harmonic electro-magnetic scattering or radiation problem governed by Maxwell’s equations, i. e., − rot H + iωεE = F in Ω, E × ν = 0 on Γ1, rot E + iωμH = G in Ω, H × ν = 0 on Γ2, where ω ∈ ℂ \ (0) and Ω ⊂ ℝ3 is an exterior weak Lipschitz domain with boundary Γ divided into two disjoint parts Γ1 and Γ2. We will present a solution theory using the framework of polynomially weighted Sobolev spaces for the rotation and divergence. For the physically interesting case ω ∈ ℝ \ (0), we will show a Fredholm alternative type result to hold using the principle of limiting absorption introduced by Eidus in the 1960s. The necessary a priori estimate and polynomial decay of eigenfunctions for the Maxwell equations will be obtained by transferring well-known results for the Helmholtz equation using a suitable decomposition of the fields E and H. The crucial point for existence is a local version of Weck’s selection theorem, also called Maxwell compactness property.
Kapitel in diesem Buch
- Frontmatter I
- Preface V
- Contents IX
- 1. The curl–div system: theory and finite element approximation 1
- 2. Darwin and higher order approximations to Maxwell’s equations in R3 45
- 3. Weck’s selection theorem: The Maxwell compactness property for bounded weak Lipschitz domains with mixed boundary conditions in arbitrary dimensions 77
- 4. Numerical analysis of the half-space matching method with Robin traces on a convex polygonal scatterer 105
- 5. Eigenvalue problems in inverse electromagnetic scattering theory 145
- 6. Maxwell eigenmodes in product domains 171
- 7. Discrete regular decompositions of tetrahedral discrete 1-forms 199
- 8. Some old and some new results in inverse obstacle scattering 259
- 9. The time-harmonic Maxwell equations with impedance boundary conditions in polyhedral domains 285
- 10. Time-harmonic electro-magnetic scattering in exterior weak Lipschitz domains with mixed boundary conditions 341
- 11. On an electro-magneto-elasto-dynamic transmission problem 383
- 12. Continuous dependence on the coefficients for a class of non-autonomous evolutionary equations 403
Kapitel in diesem Buch
- Frontmatter I
- Preface V
- Contents IX
- 1. The curl–div system: theory and finite element approximation 1
- 2. Darwin and higher order approximations to Maxwell’s equations in R3 45
- 3. Weck’s selection theorem: The Maxwell compactness property for bounded weak Lipschitz domains with mixed boundary conditions in arbitrary dimensions 77
- 4. Numerical analysis of the half-space matching method with Robin traces on a convex polygonal scatterer 105
- 5. Eigenvalue problems in inverse electromagnetic scattering theory 145
- 6. Maxwell eigenmodes in product domains 171
- 7. Discrete regular decompositions of tetrahedral discrete 1-forms 199
- 8. Some old and some new results in inverse obstacle scattering 259
- 9. The time-harmonic Maxwell equations with impedance boundary conditions in polyhedral domains 285
- 10. Time-harmonic electro-magnetic scattering in exterior weak Lipschitz domains with mixed boundary conditions 341
- 11. On an electro-magneto-elasto-dynamic transmission problem 383
- 12. Continuous dependence on the coefficients for a class of non-autonomous evolutionary equations 403
