Geodesic
Elliptic Equations with Translations of General Form in a Half-Space
Matematičeskie zametki, Tome 111 (2022) no. 4, pp. 571-580

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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.
Keywords: differential-difference equations, elliptic problems in a half-space, translations with respect to arbitrary variables. 
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     author = {A. B. Muravnik},
     title = {Elliptic {Equations} with {Translations} of {General} {Form} in a {Half-Space}},
     journal = {Matemati\v{c}eskie zametki},
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     publisher = {mathdoc},
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     number = {4},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2022_111_4_a8/}
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A. B. Muravnik. Elliptic Equations with Translations of General Form in a Half-Space. Matematičeskie zametki, Tome 111 (2022) no. 4, pp. 571-580. http://geodesic.mathdoc.fr/item/MZM_2022_111_4_a8/