The Dirichlet problem for an elliptic operator in a domain with cusp point
The paper is devoted to the study of qualitative properties of solutions to the homogeneous Dirichlet boundary value problem for a properly elliptic operator of the order \(2m\) with smooth coefficients on a plane domain \(U\) with a boundary having a cusp point in the origin \(0\). The boundary of \(U\) in a sufficiently small neighbourhood of \(0\) is described as a graph of two smooth functions \(\varphi_1, \varphi_2\) such that \[ \varphi_1, \varphi_2 \in C^1([0,\epsilon]) \cap C^{\infty}(]0,\epsilon]), \] \[ \varphi_1 < \varphi_2,\quad \varphi_1(0) = \varphi_2(0) = 0,\quad \varphi_1'(0) = \varphi_2'(0) = 0. \] The author gives conditions on \(\varphi_1, \varphi_2\) guaranteeing the smoothness of solutions or eigenfunctions of the problem and, if a solution is not smooth, he gives an asymptotic expansion near the cusp point.
scientific article; zbMATH DE number 1066075