On the stability of index of nonlocal elliptic problems

Abstract

One of the first publications devoted to nonlocal elliptic problems was the paper of T. Carleman [S]. Together with this paper are closely associated further investigations of singular integral equations with shifts and elliptic boundary value problems with shifts which map the boundary onto itself [9]. Bibliographies of publications and the theory of abstract nonlocal elliptic problems are contained in [4, 71. In [2] the Laplace equation was considered with boundary conditions connecting the values of the unknown function on a manifold r, c aQ (Q c R”) with its values on some manifold o(T, ) c Q and Dirichlet conditions on 13Q\f,. The solvability of this problem in various functional spaces was studied in [16-18,201 and elsewhere. In [ 17,201 it was shown that in the case o(F,) n (r,\r,) # 0 the Dirichlet problem with arbitrary small coefficients in nonlocal terms can have a negative index. In [lo] the formula of index for some nonlocal elliptic boundary value problems of second order in a plane domain was obtained. In the present paper we consider elliptic equations of order 2m in a bounded domain Q c R with nonlocal conditions connecting the traces of the unknown function and its derivatives on the manifolds rj (IJ, F, = 8Q) with its traces on some compacturn a c Q. In the case a c Q (Section 2) and in the case Sz n K = 0 (Section 3) we prove the stability of index of nonlocal problems in Sobolev spaces and weighted spaces, where Xi = Uj (T,\r,). In particular the set D can consist of one or several manifolds (see [2, 163). In Section 4 it is established that in the case Q n Xi = 0 index of the equation of second order, with Dirichlet conditions containing nonlocal perturbations, is equal to zero. The elliptic problems with nonlocal conditions are closely associated with boundary value problems for elliptic functional differential equations [ 171. The theory of functional differential equations for the functions of one variable was reflected in the publications most completely [IS, 131. The 323 0022-247X/91 $3.00