Solvability of a fourth order elliptic problem in a bounded sector, part I

Abstract

The purpose of this article (composed of two parts) is the study of the generalized dispersal operator of a reaction-diffusion equation in $L^p$-spaces set in the finite conical domain $S_{\omega,\rho}$ of angle $\omega>0$ and radius $\rho$ > 0 in $\mathbb{R}^2$. This first part is devoted to the behavior of the solution near the top of the cone which is completely described in the weighted Sobolev space $W^{4,p}_{3-\frac{1}{p}}(S_{\omega,\rho_0})$, $\rho_0 \leqslant \rho$, see Theorem 2.2.