Elliptic differential-difference equations with differently directed translations in half-spaces

Abstract

We study the Dirichlet problem in the half-space for elliptic differential-difference equations with operators being the compositions of differential operators and translation operators acting on spatial variables, which are independent variables ranging in the entire real axis. These equations generalize essentially the classical elliptic partial differential equations and they arise in various applications of mathematical physics, which are characterized by nonlocal and (or) inhomogeneous properties of the process or medium. In theoretical terms, an interest in such equations is due to the fact that they relate the values of the unknown function to each other (and its derivatives) not at one point, but at different points, which makes many classical methods not applicable. For the considered problem we establish the solvability in the sense of generalized functions, while for the equation a classical solvability is proved. We also find an integral representation of the solution by a Poisson type formula and we prove that the constructed solution is classical outside boundary hyperplane and uniformly tends to zero as the only independent variable, changing on the positive axis orthogonal to the boundary data hyperplane, tends to infinity. Earlier, there were studied only the cases when the translation operator acts only in one spatial variable. In this work, the translation operators act on each spatial variable. To obtain the Poisson kernel, we use classic operation scheme by Gelfand-Shilov: we apply Fourier transform to the problem with respect to all spatial variables and use the fact that the translation operators, as well as differential operators, are Fourier multipliers. Then we study the obtained Cauchy problem for the ordinary differential equation depending on dual variables as on parameters.