Spectral properties of parabolic layer potentials and transmission boundary problems in nonsmooth domains
Let \(\Omega \) be a bounded Lipschitz domain and \(T>0\). Let \(K\) be the caloric double layer potential operator on \(\partial \Omega \times (0,T)\). The authors show that there is \(\varepsilon >0\) such that \(\lambda I+K\) is invertible on \(L^p(\partial \Omega \times (0,T))\) for each \(p\in (2- \varepsilon,2+\varepsilon )\) and a real \(\lambda \) with \(| \lambda | \geq 1/2\). A similar result for harmonic double layer potentials was shown by \textit{L. Escauriaza, E. Fabes} and \textit{G. Verchota} [Proc. Am. Math. Soc. 115, 1069--1076 (1992; Zbl 0761.35013)]. This result was used for the study of the initial transmission boundary value problem for the heat equation.
scientific article; zbMATH DE number 2028876