Elliptic Equations with Translations of General Form in a Half-Space

Abstract: We study the Dirichlet problem in a half-space for elliptic differential-difference equations with operators representing superpositions of differential operators and translation operators. In each superposition, the second-derivative operator and the translation operator act with respect to arbitrary independent tangential (space-like) variables. For this problem, solvability in the sense of generalized functions (distributions) is established, an integral representation of the solution is constructed by means of a Poisson-type formula, its infinite smoothness outside the boundary hyperplane is proved, and its convergence to zero (together with all of its derivatives) as the time-like independent variable tends to infinity is established.