Asymptotics of Solutions of the Poisson Equation near Straight Edges

Abstract

The paper deals with the Dirichlet 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 author describes the singularities of the Green function near the edge of the domain. Using this result, he 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. Here, precise formulas for all coefficients in the asymptotics are given.