Well-posedness theory for electromagnetic obstacle problems

Irwin Yousept

doi:10.1016/j.jde.2020.05.009

Abstract

This paper develops a well-posedness theory for hyperbolic Maxwell obstacle problems generalizing the result by Duvaut and Lions (1976) [5]. Building on the recently developed result by Yousept (2020) [30], we prove an existence result and study the uniqueness through a local H(curl)-regularity analysis with respect to the constraint set. More precisely, every solution is shown to locally satisfy the Maxwell-Ampère equation (resp. Faraday equation) in the region where no obstacle is applied to the electric field (resp. magnetic field). By this property, along with a structural assumption on the feasible set, we are able to localize the obstacle problem to the underlying constraint regions. In particular, the resulting localized problem does not employ the electric test function (resp. magnetic test function) in the area where the L2-regularity of the rotation of the electric field (resp. magnetic field) is not a priori guaranteed. This localization strategy is the main ingredient for our uniqueness proof. After establishing the well-posedness, we consider the case where the electric permittivity is negligibly small in the electric constraint region and investigate the corresponding eddy current obstacle problem. Invoking the localization strategy, we derive an existence result under an L2-boundedness assumption for the electric constraint region along with a compatibility assumption on the initial data. The developed theoretical results find applications in electromagnetic shielding.

Highlights

More than four decades ago, Duvaut and Lions [5, Chapter 7, Section 8] proposed and analyzed a Maxwell obstacle problem describing the propagation of electromagnetic waves in a polarizable medium with an obstacle constraint on the electric field of the form
We developed an existence and uniqueness theory for the electromagnetic obstacle problem (P)
The general one is based on a localization strategy, leading to a localized variational inequality (3.15) on the electric and magnetic constraint regions c E

Summary

Introduction

More than four decades ago, Duvaut and Lions [5, Chapter 7, Section 8] proposed and analyzed a (hyperbolic) Maxwell obstacle problem describing the propagation of electromagnetic waves in a polarizable medium with an obstacle constraint on the electric field of the form. Along with the existence result, we show that every solution locally satisfies (i) the Maxwell-Ampère equation in the electric free region E, (ii) the Faraday equation in the magnetic free region H. On this basis, the uniqueness analysis is studied. In use of Assumption 1.1, particular, the resulting we are able to localize localized problem does (P) not into the employ constraint regions the magnetic test c E and function The final part of this paper considers the case where the electric permittivity is negligibly small in problem the electric constraint (1.19) by neglecting inregicEonandcEd.eWrivee investigate the resulting eddy current obstacle an existence result for (1.19) (Theorem 3).

Problem formulation and main results

Preliminaries

Proof of Theorem 1

Proof of Theorem 2

Proof of Theorem 3

Conclusion