The Zaremba Problem with Singular Interfaces as a Corner Boundary Value Problem

Abstract

We study mixed boundary value problems for an elliptic operator $A$ on a manifold $X$ with boundary $Y$, i.e., $A\,u=f$ in $\mathrm{int}\,X, T_{\pm}u=g_{\pm}$ on $\mathrm{int}\,Y_{\pm}$, where $Y$ is subdivided into subsets $Y_{\pm}$ with an interface $Z$ and boundary conditions $T_{\pm}$ on $Y_{\pm}$ that are Shapiro–Lopatinskij elliptic up to $Z$ from the respective sides. We assume that $Z\subset Y$ is a manifold with conical singularity $v$. As an example we consider the Zaremba problem, where $A$ is the Laplacian and $T_{-}$ Dirichlet, $T_{+}$ Neumann conditions. The problem is treated as a corner boundary value problem near $v$ which is the new point and the main difficulty in this paper. Outside $v$ the problem belongs to the edge calculus as is shown in Bull. Sci. Math. (to appear).

With a mixed problem we associate Fredholm operators in weighted corner Sobolev spaces with double weights, under suitable edge conditions along $Z\setminus \{v\}$ of trace and potential type. We construct parametrices within the calculus and establish the regularity of solutions.