Spectral properties of the layer potentials on Lipschitz domains
Let \(K\) be the double layer potential related to the Laplace equation on a bounded Lipschitz domain \(\Omega\) in \({\mathbb R}^n\), \(n\geq 2\), and let \(K^*\) be the adjoint of \(K\). The authors study some spectral properties of \(K^*\). In particular, they show that when \(\Omega\) is a locally convex bounded Lipschitz domain, then \(\beta I-K^*\) is invertible in \(H^{-\alpha}(\partial \Omega)\), \(\alpha\in [0, 1]\), for all complex numbers \(\beta\) with \(|\beta|>\frac12\), and that when \(\Omega\) is an arbitrary bounded Lipschtitz domain, then \(\beta I-K^*\) is invertible in \(H^{-1/2}(\partial \Omega)\) for all \(\beta\in {\mathbb C}\setminus (-\frac12, \frac12]\).
scientific article; zbMATH DE number 5615648