Behavior of Solutions of the Neumann Problem for the Poisson Equation near Straight Edges

Abstract

The paper deals with the Neumann problem for the Poisson equation $\Delta u=f$ in the domain $\mathcal{D}=K\times {\mathbb{R}}^{n-m}$, where $K$ is a cone in ${\mathbb{R}}^m$. The first part of the paper is concerned with the singularities of the Green function near the edge of the domain. Using the decomposition of the Green function given in the first part, the author obtains the asymptotics of the solution of the boundary value problem for a right-hand side $f$ belonging to a weighted $L_p$ Sobolev space. Precise formulas for all coefficients in the asymptotics are given.