Regularity results for Laplace interface problems in two dimensions (Q5945535)

This paper is devoted to the Laplace interface problems on a domain \(\Omega\subset\mathbb R^2\). The strong form of the problem is:NEWLINENEWLINE\[NEWLINE\nabla\cdot k(x)\nabla u(x)=f(x), \quad x\in\Omega,NEWLINE\]NEWLINENEWLINEwhere the coefficient \(k\) is bounded byNEWLINENEWLINE\[NEWLINE\delta\leq k(x)\leq\delta^{-1}, \quad x\in\Omega,NEWLINE\]NEWLINENEWLINEfor some \(\delta>0\) and where mixed boundary conditions are imposed. Here the author discusses piecewise \(H^s\)-regularity of interface problems for the Laplacian which holds independently of the number and shape of the subdomains on which the coefficient \(k\) is constant. In contrast to many previous results on \(H^s\) regularity, in this paper there are no restrictions on the maximal number of subdomains which share a point. The main result states that quasi-monotonicity is necessary and sufficient to have \(H^{1,\frac14}\)-regularity independently of the global bounds of \(k\) and without restrictions on the subdomains.