Shape Holomorphy and Sparse Approximation of the Maxwell Electric Field Integral Operator
Uncertainty quantification for time-harmonic Maxwell scattering by obstacles of uncertain shape needs more than holomorphic dependence of the scattered field: for a boundary element method it is the boundary integral operator family itself that must depend holomorphically on the shape parameters. Two obstructions stand in the way. The natural energy space of the electric field integral equation, $\boldsymbol H^{-1/2}_{\mathrm{div}_Γ}(Γ)$, depends on the geometry, and the available operator-valued shape-holomorphy theory for weakly singular kernels is set in $L^2$, which does not reach it. We remove both. A surface contravariant Piola transformation identifies the geometry-dependent Maxwell trace spaces with a fixed reference space, and in the pulled-back variational formulation the surface Jacobians cancel exactly. The principal analytical ingredient is then a uniform fractional mapping theorem $H^{-1/2}\to H^{1/2}$ for the complex-deformed scalar single-layer family on uniformly $C^{1,1}$ surfaces, obtained by realizing the Laplace principal part as the trace of a complex-coefficient Newton problem on a fixed ambient space. The pulled-back operators are consequently $(\bm b,p,\eps)$-holomorphic for $\bm b\in\ell^p(\N)$, $0<p<1$, and pointwise exclusion of interior electric resonances over the compact real parameter set yields uniform invertibility. Legendre coefficients are therefore $\ell^p$ summable, so the operator family, the surface current and the far field all admit sparse polynomial approximations at dimension-independent best $N$-term rates. These statements are for the operator family itself in its energy-space operator norm, not only for individual solutions.