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.

中文翻译：

---

半空间中的椭圆微分方程

考虑了椭圆微分方程的半空间中的狄利克雷问题，该方程的算子是微分和差分算子的组合。对于这个问题，证明了经典的可解性或几乎处处的可解性（取决于对边界数据施加的约束），根据泊松型公式构建已找到解的积分表示，并将其收敛到零作为时间证明自变量趋于无穷大。