THE DISCRETE DIRICHLET PROBLEM. SOLVABILITY AND APPROXIMATION PROPERTIES

We consider the discrete Dirichlet boundary value problem for a discrete elliptic pseudodifferential equation in the quadrant and study its solvability in discrete counterparts of the Sobolev-Slobodetskii space. The study is based on a special factorization of the elliptic symbol. We compare the solutions to the discrete Dirichlet problem and its continuous counterpart. Bibliography: 10 titles. In this paper, based on the ideas and methods of [1, 2] (cf. also [3] [7]), we compare discrete and continuous elliptic boundary value problems in the quadrant for the simplest pseudodifferential operators. We emphasize that, in the case of a quadrant, there are principal differences from the case of a half-space, and new analytic tools are required.
1 Preliminaries We recall the main notions and results which will be used throughout the paper (we refer to [5] for details). Let 1? be an integer lattice in the [...]