NEUMANN LAPLACIAN ON A DOMAIN WITH TANGENTIAL COMPONENTS IN THE BOUNDARY
С. А. Назаров, Jan Sokołowski, Jari Taskinen · 2008
Abstract. We study the Neumann problem for the Poisson equation in a domain where two boundary components are tangential at a single point, such that a geometrical irregularity of the rotational cusp type is formed. We derive necessary and sucient conditions for the existence of a solution with a nite Dirichlet integral. 1. Formulation of the problem. Since the pioneering paper [2] by V.Kondratiev much attention is devoted to elliptic boundary value problems in domains with irregular boundaries. An almost complete theory has been created in case of conical corner points and edges, see for example the key works [2], [9], [10], [16] and [3], [23], [11], [12], respectively, and also the monographs [1], [18], [4]. Other geometrical irregularities, common in everyday life and engineering practise, have been considered as well. The papers [8], [24] and [4] contain studies near cusp tips of peak{shaped domains, using reduction to conical domains and to dierential operators with strongly perturbed coecients. There are nevertheless many types of irregularities, also interesting for applied