Elliptic differential-difference problems in half-spaces: case of summable functions

Abstract

We study the Dirichlet problem in the half-space for elliptic equations involving, apart of differential operators, the shift operators acting in tangential (spatial-like) variables, that is, in independent variables varying in entire real line. The boundary function in the problem is supposed to be summable, which in the classical case corresponds to the situation, in which only solutions with finite energy are possible. We consider two principally different cases: the case, in which the studied equation involves superpositions of differential operators and the shift operators and the case, when it involves their sums, that is, it is an equation with nonlocal potentials. For both types of problems we construct an integral representation of the solution to this problem in the sense of generalized functions and we prove that its infinitely smoothness in an open half-space (i.e., outside the boundary hyperplane) and tends uniformly to zero together with all its derivatives as a time-like variable tends to infinity; this time-like variable is a single independent variable varying on the positive half-axis. The rate of this decay is power-law; the degree is equal to the sum of the dimension of the space-like independent variable and the order of the derivative of the solution. The most general current results are presented: shifts of independent variables are allowed in arbitrary (tangential) directions, and if there are several shifts, no conditions of commensurability are imposed on their values. Thus, just as in the classical case, problems with summable boundary functions fundamentally differ from the previously studied problems with essentially bounded boundary functions: the latter, as previously established, admit solutions having no limit when a time-like variable tends to infinity, and the presence or absence of such a limit is determined by the Repnikov-Eidelman stabilization condition.