Regularity Results for Laplace Interface Problems in Two Dimensions

Abstract

We investigate the regularity of solutions of interface problems for the Laplacian in two dimensions. Our objective are regularity results which are independent of global bounds of the data (the diffusion). Therefore we use a restriction on the data, the quasi-monotonicity condition, which we show to be sufficient and necessary to provide $H^{1+\frac{1}{4}}$-regularity. In the proof we use estimates of eigenvalues of a related Sturm-Liouville eigenvalue problem. Additionally we state regularity results depending on the data.