Transmission problems for Maxwell's equations with weakly Lipschitz interfaces

Abstract

SUMMARYWe prove su cient conditions on material constants, frequency and Lipschitz regularity of interface for well posedness of a generalized Maxwell transmission problem in ÿnite energy norms. This is done by embedding Maxwell's equations in an elliptic Dirac equation, by constructing the natural trace space for the transmission problem and using Hodge decompositions for operators d and on weakly Lipschitz domains to prove stability. We also obtain results for boundary value problems and transmission problems for the Hodge-Dirac equation and prove spectral estimates for boundary singular integral operators related to double layer potentials.

Solvability of elliptic systems with square integrable boundary data

“…The condition (5) lies between pointwise strict accretivity, i.e. (7) Re(A(x)v, v) ≥ˇ|v| 2 for all v ∈ C (1+n)m and a.e. x ∈ R n , and the R n+1 + Gårding inequality…”

“…[7] for explanations of this terminology.) When A=A 0 , then as in Section 4.2 it follows that K A0 =0, since the diagonal entries of sgn(DÂ) are zero in the normal/tangential splitting of the space.…”

Hilbert Transforms on the Sphere with the Clifford Algebra Setting

“…In a meeting in Finland he first explained to the first author the definition of Hilbert transform on curves and surfaces that is closely related to his Ph.D. thesis work [1]. The idea of the present work is under heavy influence of Axelsson.…”

“…In [3] we further extend the results of the present paper to harmonic k-forms in Lipschitz domains. For harmonic conjugates of the Hodge type in general we refer the reader to Axelsson's paper [2] and thesis [1]. The present study proposes a systematic approach based merely on the Cauchy integral, via Plemelj-Sokhotzki Theorem in the context.…”

Spectral theory and iterative methods for the Maxwell system in nonsmooth domains

“…Related results for singular integral operators associated with the Laplacian in Lipschitz domains can be found in [25]. Also [1] includes a study of the L 2 spectrum of the so-called rotation singular integral operator, introduced in connection to the study of transmission problems for Maxwell's equations with Lipschitz interfaces.…”