On Ventcel’s type boundary condition for Laplace operator in a sector. (English) Zbl 1291.35048

Popivanov, Petar; Slavova, Angela

J. Geom. Symmetry Phys. 31, 119-130 (2013).

Summary: This paper deals with classical solutions of the Dirichlet-Ventcel boundary value problem (BVP) for the Laplace operator in a bounded sector in the plane having opening of the corresponding angle \(\varphi_0>0\). Ventcel BVP is given by a second order differential operator on the boundary satisfying the 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 a uniqueness result via the maximum principle and then the 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.

MSC:

35J25	Boundary value problems for second-order elliptic equations
35A01	Existence problems for PDEs: global existence, local existence, non-existence
35A02	Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35J08	Green’s functions for elliptic equations