IRMAR, Université de Rennes 1
On the regularity of electromagnetic fields on Lipschitz domains
The usual variational spaces \(X_{N}\) and \(X_{T}\) 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 \(H^{1/2}(\Omega)\). If the domain is, in addition, piecewise smooth, then there exists always an \(\varepsilon>0\) such that these spaces are contained in \(H^{1/2+\varepsilon}(\Omega)\), 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 \(\Omega\) that is even of class \(C^{1}\), for which such an \(\varepsilon>0\) does not exist, that is, the spaces
\(
X_{N} = H(\mathrm{div},\Omega)\cap H_{0}(\mathrm{curl},\Omega) \quad\mbox{ and }
X_{T} = H_{0}(\mathrm{div},\Omega)\cap H(\mathrm{curl},\Omega)
\)
are contained in \(H^{s}(\Omega)\) for \(s=1/2\), but not for any \(s>1/2\). The construction uses ideas from N.~Filonov’s construction of a \(C^{3/2}\) domain for which the usual Birman-Solomyak decomposition of \(X_{T}\) is not possible.