Dirk Pauly

Maxwell meets Korn: A new coercive inequality for tensor fields in RN×N with square-integrable exterior derivative

For a bounded domain with connected Lipschitz boundary, we prove the existence of some c > 0, such that holds for all square-integrable tensor fields , having square-integrable generalized “rotation” tensor fields and vanishing tangential trace on ∂Ω, where both operations are to be understood row-wise. Here, in each row, the operator curl is the vector analytical reincarnation of the exterior derivative d in . For compatible tensor fields T, that is, T = ∇ v, the latter estimate reduces to a non-standard variant of Korns first inequality in , namely for all vector fields , for which ∇ vn,n = 1, … ,N, are normal at ∂Ω. Copyright

Hodge–Helmholtz decompositions of weighted Sobolev spaces in irregular exterior domains with inhomogeneous and anisotropic media

We study in detail Hodge–Helmholtz decompositions in nonsmooth exterior domains Ω⊂ℝN filled with inhomogeneous and anisotropic media. We show decompositions of alternating differential forms of rank q belonging to the weighted L2-space Ls2, q(Ω), s∈ℝ, into irrotational and solenoidal q-forms. These decompositions are essential tools, for example, in electro-magnetic theory for exterior domains. To the best of our knowledge, these decompositions in exterior domains with nonsmooth boundaries and inhomogeneous and anisotropic media are fully new results. In the Appendix, we translate our results to the classical framework of vector analysis N=3 and q=1, 2. Copyright

Complete low frequency asymptotics for time-harmonic generalized Maxwell equations in nonsmooth exterior domains

We discuss the radiation problem of total reflection for a time-harmonic generalized Maxwell system in a non-smooth exterior domain with non-smooth inhomogeneous, anisotropic coefficients converging near infinity with a certain rate towards the identity. By means of the limiting absorption principle, a Fredholm alternative holds true and the eigensolutions decay polynomially resp. exponentially at infinity. We prove that the corresponding eigenvalues do not accumulate even at zero. Next, we show the convergence of the time-harmonic solutions to a solution of an electro-magneto static Maxwell system as the frequency tends to zero. Finally we are able to generalize these results easily to the corresponding Maxwell system with inhomogeneous boundary data. This paper is thought of as the first and introductory one in a series of three papers, which will completely discuss the low frequency behavior of the solutions of the time-harmonic Maxwell equations.

Generalized electro-magneto statics in nonsmooth exterior domains

Summary We develop a solution theory for a generalized electro-magneto static Maxwell system in an exterior domain Ω ⊂ RN with anisotropic coefficients converging at infinity with a rate r-τ, τ > 0, towards the identity. Our main goal is to treat right hand side data from some polynomially weighted Sobolev spaces and obtain solutions which are up to a finite sum of special generalized spherical harmonics in another appropriately weighted Sobolev space. As a byproduct we prove a generalized spherical harmonics expansion suited for Maxwell equations. In particular, our solution theory will allow us to give meaning to higher powers of a special static solution operator. Finally we show, how this weighted static solution theory can be extended to handle inhomogeneous boundary data as well. This paper is the second one in a series of three papers, which will completely reveal the low frequency behavior of solutions of the time-harmonic Maxwell equations.

Regularity results for generalized electro-magnetic problems

Abstract We prove regularity results up to the boundary for time independent generalized Maxwell equations on Riemannian manifolds with boundary using the calculus of alternating differential forms. We discuss homogeneous and inhomogeneous boundary data and show ‘polynomially weighted regularity in exterior domains as well’.

ON KORN'S FIRST INEQUALITY FOR TANGENTIAL OR NORMAL BOUNDARY CONDITIONS WITH EXPLICIT CONSTANTS

We will prove that for piecewise C 2 -concave domains in R N Korns rst inequality holds for vector elds satisfying homogeneous normal or tangential boundary conditions with explicit Korn constant p 2.

On polynomial and exponential decay of eigen-solutions to exterior boundary value problems for the generalized time-harmonic Maxwell system

We prove polynomial and exponential decay at infinity of eigen-vectors of partial differential operators related to radiation problems for time-harmonic generalized Maxwell systems in an exterior domain Ω ⊂ R , N 1, with non-smooth inhomogeneous, anisotropic coefficients converging near infinity with a rate r−τ , τ > 1, towards the identity. As a canonical application we show that the corresponding eigen-values do not accumulate and that by means of Eidus’ limiting absorption principle a Fredholm alternative holds true.

A note on the justification of the eddy current model in electrodynamics

The issue of justifying the eddy current approximation of Maxwells equations is re-considered in the time-dependent setting. Convergence of the solution operators is shown in the sense of strong operator limits.

An Elementary Method of Deriving A Posteriori Error Equalities and Estimates for Linear Partial Differential Equations

Abstract In this paper we present a simple method of deriving a posteriori error equalities and estimates for linear elliptic and parabolic partial differential equations. The error is measured in a combined norm taking into account both the primal and dual variables. We work only on the continuous (often called functional) level and do not suppose any specific properties of numerical methods and discretizations.

Functional A Posteriori Error Control for Conforming Mixed Approximationsof Coercive Problems with Lower Order Terms

Abstract The results of this contribution are derived in the framework of functional type a posteriori error estimates. The error is measured in a combined norm which takes into account both the primal and dual variables denoted by x and y, respectively. Our first main result is an error equality for all equations of the class A * ⁢ A ⁢ x + x = f