Elliptic Differential-Difference Equations with Nonlocal Potentials in a Half-Space

Differential-difference equations (and functional-differential equations overall) find applications in areas that are not covered by classical models of mathematical physics, namely, in models of nonlinear optics, in nonclassical diffusion models (taking into account the inertial nature of this physical phenomenon), in biomathematical applications, and in the theory of multilayered plates and shells. This is explained by the nonlocal nature of functional-differential equations: unlike in classical differential equations, where all derivatives of the unknown function (including the function itself) are related at the same point (which is a reduction of the mathematical model), in functional-differential equations, these terms can be related at different points, thus substantially expanding the generality of the model. This paper deals with the Dirichlet problem in a half-space for elliptic differential-difference equations with nonlocal potentials: the differential operators act on the unknown (sought) function at one point, while the potential, at another. For integrable boundary data (in which case only finite-energy solutions are admissible), an integral representation of the solution is constructed and its smoothness outside the boundary hyperplane is proved. Moreover, this representation is shown to tend uniformly to zero as the timelike variable increases unboundedly.