On Ventcel's Type Boundary Condition for Laplace Operator in a Sector

Abstract

This paper deals with classical solutions of the Dirichlet-Ventcel boundary value problem (BVP) for the Laplace operator in bounded sector in the plane having opening of the corresponding angle $ \varphi_{0} > 0 $. Ventcel BVP is given by second order differential operator on the boundary satisfying Lopatinksii condition there. As the boundary is non smooth, two different cases appear: $ \frac{\pi}{\varphi_{0}} $ is irrational and $ \frac{\pi}{\varphi_{0}} $ is an integer. At first we prove uniqueness result via the maximum principle and then existence of the classical solution. To do this we apply two different approaches: the machinery of the small denominators and the concept of Green function.