Elliptic Differential-Difference Equations in the Half-Space

The Dirichlet problem in the half-space for elliptic differential-difference equations with operators that are compositions of differential and difference operators is considered. For this problem, classical solvability or solvability almost everywhere (depending on the constraints imposed on the boundary data) is proved, an integral representation of the found solution in terms of a Poisson-type formula is constructed, and its convergence to zero as the time-like independent variable tends to infinity is proved.