Elliptic Equations with General-Kind Nonlocal Potentials in Half-Spaces

In this paper, the Dirichlet problem in half-spaces is investigated for elliptic differential-difference equations such that the potential admits translations in arbitrary directions. Such equations with nonlocal potentials arise in various applications (not covered by classical differential equations of mathematical physics), while elliptic problems in anisotropic domains represent an independent research interest because phenomena specific for nonstationary equations frequently arise in such cases. We construct integral representations of solutions (expressed by convolutions of boundary-value functions with a Poisson-like kernel), prove its infinite smoothness outside the boundary hyperplane, and prove its uniform power-like decay as the timelike independent variable tends to infinity.