Martin Costabel

IRMAR, Université de Rennes 1

On the regularity of electromagnetic fields on Lipschitz domains

The usual variational spaces Latex formula and Latex formula for time-harmonic electric or magnetic fields with PEC boundary conditions on a bounded Lipschitz domain have long been known to be contained in the Sobolev space Latex formula. If the domain is, in addition, piecewise smooth, then there exists always an Latex formula such that these spaces are contained in Latex formula, and this additional smoothness is known to be useful in the analysis of some numerical algorithms. The question has recently come up whether such an additional regularity is also present for every Lipschitz domain. In the talk, I will describe the construction of a domain Latex formula that is even of class Latex formula, for which such an Latex formula does not exist, that is, the spaces


Latex formula

are contained in Latex formula for Latex formula, but not for any Latex formula. The construction uses ideas from N.~Filonov’s construction of a Latex formula domain for which the usual Birman-Solomyak decomposition of Latex formula is not possible.

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