On the Dirichlet and Neumann problems in multi-dimensional cone

Abstract

We consider an elliptic pseudodifferential equation in a multi-dimensional cone, and using the wave factorization concept for an elliptic symbol we describe a general solution of such equation in Sobolev-Slobodetskii spaces. This general solution depends on some arbitrary functions, their quantity being determined by an index of the wave factorization. For identifying these arbitrary functions one needs some additional conditions, for example, boundary conditions. Simple boundary value problems, related to Dirichlet and Neumann boundary conditions, are considered. A certain integral representation for this case is given.