Die Maxwellgleichung mit wechselnden Randbedingungen (The Maxwell Equation with Mixed Boundary Conditions)

TL;DR

This thesis provides a comprehensive mathematical analysis of Maxwell's equations with mixed boundary conditions on Riemannian manifolds, including compactness, decompositions, estimates, and solution theories.

Contribution

It introduces new results on trace, extension, and regularity theorems, and develops a detailed solution framework for static Maxwell problems with mixed boundary conditions.

Findings

Proves compactness and Hodge decompositions for Maxwell equations.

Establishes trace and extension theorems for boundary conditions.

Develops a solution theory for static Maxwell problems.

Abstract

In the thesis at hand we give a comprehensive discussion of basic problems for generalized Maxwell equations with mixed boundary conditions using the calculus of alternating differential forms on Riemannian manifolds of arbitrary dimension. We prove compactness results, Hodge decompositions and Poincare type estimates. For the case of 'full' boundary conditions we present trace and extension theorems, regularity theory as well as a detailed solution theory for static Maxwell problems.