Shape Holomorphy and Sparse Approximation of the Maxwell Electric Field Integral Operator
Abstract
Time-harmonic Maxwell scattering by a perfectly conducting obstacle is governed by the electric field integral equation (EFIE), whose natural energy space is H−1/2(divΓ,Γ). For countably parametrized surface deformations, both the operator and its energy space depend on the geometry. We establish operator-valued shape holomorphy of the EFIE operator on a fixed reference space and derive dimension-independent sparse approximation rates. The main analytical difficulty is a fractional mapping property absent from existing L2-based operator-valued shape-holomorphy theory for weakly singular kernels: the Maxwell graph norm requires uniform holomorphy of the complex-deformed scalar single layer as an operator from H−1/2 to H1/2. We prove this one-order smoothing by realizing the Laplace principal part as the trace of a uniformly sectorial complex-coefficient divergence-form problem on a fixed real ambient domain. Combined with a surface contravariant Piola transformation, which identifies the geometry-dependent Maxwell trace spaces and cancels the surface Jacobians in the pulled-back EFIE exactly, this yields (b,p,ε)-holomorphy of the EFIE operator family for ℓp-summable shape deformations, 0<p<1. Under pointwise exclusion of interior electric resonances, parameter-uniform invertibility is derived rather than assumed, and the surface current and far field inherit the same parametric regularity. Their Legendre coefficients are ℓp-summable, yielding dimension-independent best N-term approximation rates. The operator-level results also transfer to Galerkin boundary element operators on a single reference mesh, with constants independent of the discretization dimension, and provide reusable surrogates for multiple incident fields and bounded linear observables.
Related
- Numerical Approximation of Maxwell Transmission Eigenvalue Problem in Dispersive Media via Holomorphic Operator Function
- Shape Holomorphy of the Helmholtz Calderón Projector and the Maxwell EFIE on C1,1 Surfaces and Screens
- The Calderón operator for time-harmonic Maxwell's equations in closed waveguides
- High-Order-Accurate Continuity Enforcing Nyström Discretization of 3D Maxwell Combined Field Integral Equations
- Operator-Theoretic Stability and Observer Synthesis for Parameter-Dependent Vlasov--Maxwell Dynamics
BibTeX
@misc{escapilinchauspe2026shape,
title = {{Shape Holomorphy and Sparse Approximation of the Maxwell Electric Field Integral Operator}},
author = {Paul Escapil-Inchauspé and Carlos Jerez-Hanckes},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00466},
eprint = {2609.00466},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00466},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Shape Holomorphy and Sparse Approximation of the Maxwell Electric Field Integral Operator
AU - Paul Escapil-Inchauspé
AU - Carlos Jerez-Hanckes
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.00466
UR - https://arxiv.org/abs/2609.00466
AB - Time-harmonic Maxwell scattering by a perfectly conducting obstacle is governed by the electric field integral equation (EFIE), whose natural energy space is H^−1/2(div_Γ,Γ). For countably parametrized surface deformations, both the operator and its energy space depend on the geometry. We establish operator-valued shape holomorphy of the EFIE operator on a fixed reference space and derive dimension-independent sparse approximation rates. The main analytical difficulty is a fractional mapping property absent from existing L^2-based operator-valued shape-holomorphy theory for weakly singular kernels: the Maxwell graph norm requires uniform holomorphy of the complex-deformed scalar single layer as an operator from H^−1/2 to H^1/2. We prove this one-order smoothing by realizing the Laplace principal part as the trace of a uniformly sectorial complex-coefficient divergence-form problem on a fixed real ambient domain. Combined with a surface contravariant Piola transformation, which identifies the geometry-dependent Maxwell trace spaces and cancels the surface Jacobians in the pulled-back EFIE exactly, this yields (b,p,ε)-holomorphy of the EFIE operator family for ℓ^p-summable shape deformations, 0<p<1. Under pointwise exclusion of interior electric resonances, parameter-uniform invertibility is derived rather than assumed, and the surface current and far field inherit the same parametric regularity. Their Legendre coefficients are ℓ^p-summable, yielding dimension-independent best N-term approximation rates. The operator-level results also transfer to Galerkin boundary element operators on a single reference mesh, with constants independent of the discretization dimension, and provide reusable surrogates for multiple incident fields and bounded linear observables.
ER -