Boundary Value Problems for the Laplace Operator in an Infinite Plane Sector

Abstract

The starting point of the present paper (and some previous papers) is the theory of boundary value problems for elliptic pseudodifferential equations in domains with smooth boundary, constructed in [1, 2] in the mid-sixties. Nearly at the same time, the paper [3] was published, where the normal solvability was analyzed for general boundary value problems for differential operators in domains with conical (or corner) points (see also [4]). More complicated singularities like the vertex of a polyhedral angle and edges of various dimensions were also considered (V.G. Maz’ya, B.A. Plamenevskii, and A.I. Komech). Obviously, these papers do not exhaust the entire variety of problems appearing in nonsmooth situations. Very recently, the papers [5, 6] have appeared, where boundary value problems were considered for the Laplacian in a plane sector with the third boundary condition on the boundary. More complicated multidimensional statements can be found in [7, 8]. Interesting (and very close) results were obtained in [9], but only for differential operators. The author [10, 11] suggested a new approach to the theory of boundary value problems for elliptic pseudodifferential equations, which generalizes the Vishik–Eskin method to the case of nonsmooth domains and is based on a special factorization of the elliptic symbol of a pseudodifferential operator. On the other hand, this approach involves also the ideas due to Kondrat’ev and Eskin, where one applies the Mellin transform to the boundary value problem at the very beginning. This approach has the advantage that one can write out the general solution of a pseudodifferential operator in an infinite plane sector and use it to state a boundary value problem, which can be solved in closed form in a number of cases. This will be illustrated in the present paper for the simplest example of the Laplace operator.