Abstract:
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.
Citation:
A. B. Muravnik, “Elliptic Equations with Translations of General Form in a Half-Space”, Mat. Zametki, 111:4 (2022), 571–580; Math. Notes, 111:4 (2022), 587–594
\Bibitem{Mur22}
\by A.~B.~Muravnik
\paper Elliptic Equations with Translations of General Form in a Half-Space
\jour Mat. Zametki
\yr 2022
\vol 111
\issue 4
\pages 571--580
\mathnet{http://mi.mathnet.ru/mzm13369}
\crossref{https://doi.org/10.4213/mzm13369}
\transl
\jour Math. Notes
\yr 2022
\vol 111
\issue 4
\pages 587--594
\crossref{https://doi.org/10.1134/S0001434622030270}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85128968110}
Linking options:
https://www.mathnet.ru/eng/mzm13369
https://doi.org/10.4213/mzm13369
https://www.mathnet.ru/eng/mzm/v111/i4/p571
This publication is cited in the following 8 articles:
Vladimir Vasilyev, Natalya Zaitseva, “On Hyperbolic Equations with a Translation Operator in Lowest Derivatives”, Mathematics, 12:12 (2024), 1896
Viktoriia V. Liiko, Andrey B. Muravnik, “Elliptic equations with arbitrarily directed translations in half-spaces”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 43 (2023), 64–77
N. V. Zaitseva, A. B. Muravnik, “Klassicheskie resheniya giperbolicheskogo differentsialno-raznostnogo uravneniya so sdvigom na proizvolnyi vektor”, Izv. vuzov. Matem., 2023, no. 5, 34–40
A. B. Muravnik, N. V. Zaitseva, “Classical solutions of hyperbolic differential-difference equations with differently directed translations”, Lobachevskii J, Math., 44:3 (2023), 920
N. V. Zaitseva, “Family of smooth solutions of hyperbolic differential-difference equation”, Differential Equations, Mathematical Modeling and Computational Algorithms, Springer Proceedings in Mathematics & Statistics, 423, 2023, 289–298
V. Vasilyev, N. Zaitseva, “Classical solutions of hyperbolic equation with translation operators in free terms”, Mathematics, 11:14 (2023), 3137
N. V. Zaitseva, A. B. Muravnik, “Smooth solutions of hyperbolic equations with translation by an arbitrary vector in the free term”, Diff Equat, 59:3 (2023), 371
N. V. Zaitseva, A. B. Muravnik, “A Classical Solution to a Hyperbolic Differential-Difference Equation with a Translation by an Arbitrary Vector”, Russ Math., 67:5 (2023), 29
Citation in format AMSBIB
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