Green’s formula for elliptic operators with a shift and its applications

Abstract

General boundary value problems with a shift for elliptic equations were first studied in [1]. Such problems arise for instance when studying certain steady-state collations. In his paper we introduce the notion of normal boundary conditions with a shift and deduce the Green’s formula for such boundary conditions and partial different equations of even order. We obtain a number of applications of this formula. Namely, we introduce the notion of the adjoint problem and prove that the adjoint problem is elhptic if and only if the given problem is elliptic. We give solvability conditions for both the given and the adjoint problem in positive spaces of Sobolev type. They allow to prove isomorphism theorems (i.e., solvability theorems in complete scale of spaces) and theorems on the local increasing of moothnes, In addition we prove the existence and study the smoothness properties of the Green’s function for the problem with a shift. We investigate also the approximation of functions on a manifold by solutions of the problem with a shift.