Differential-Difference Elliptic Equations with Nonlocal Potentials in Half-Spaces

Andrey B. Muravnik ()
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Andrey B. Muravnik: Nikol’skii Mathematical Institute, Peoples Friendship University of Russia, Miklukho–Maklaya ul. 6, 117198 Moscow, Russia

Mathematics, 2023, vol. 11, issue 12, 1-9

Abstract: We investigate the half-space Dirichlet problem with summable boundary-value functions for an elliptic equation with an arbitrary amount of potentials undergoing translations in arbitrary directions. In the classical case of partial differential equations, the half-space Dirichlet problem for elliptic equations attracts great interest from researchers due to the following phenomenon: the solutions acquire qualitative properties specific for nonstationary (more exactly, parabolic) equations. In this paper, such a phenomenon is studied for nonlocal generalizations of elliptic differential equations, more exactly, for elliptic differential-difference equations with nonlocal potentials arising in various applications not covered by the classical theory. We find a Poisson-like kernel such that its convolution with the boundary-value function satisfies the investigated problem, prove that the constructed solution is infinitely smooth outside the boundary hyperplane, and prove its uniform power-like decay as the timelike independent variable tends to infinity.

Keywords: differential-difference equations; nonlocal potential elliptic equations; half-space Dirichlet problem; summable boundary-value functions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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