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

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{MZM_2022_111_4_a8,
     author = {A. B. Muravnik},
     title = {Elliptic {Equations} with {Translations} of {General} {Form} in a {Half-Space}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {571--580},
     publisher = {mathdoc},
     volume = {111},
     number = {4},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2022_111_4_a8/}
}
TY  - JOUR
AU  - A. B. Muravnik
TI  - Elliptic Equations with Translations of General Form in a Half-Space
JO  - Matematičeskie zametki
PY  - 2022
SP  - 571
EP  - 580
VL  - 111
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2022_111_4_a8/
LA  - ru
ID  - MZM_2022_111_4_a8
ER  - 
%0 Journal Article
%A A. B. Muravnik
%T Elliptic Equations with Translations of General Form in a Half-Space
%J Matematičeskie zametki
%D 2022
%P 571-580
%V 111
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2022_111_4_a8/
%G ru
%F MZM_2022_111_4_a8
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/

[1] M. A. Vorontsov, N. G. Iroshnikov, R. L. Abernathy, “Diffractive patterns in a nonlinear optical two-dimensional feedback system with field rotation”, Chaos, Solitons, and Fractals, 4:8–9 (1994), 1701–1716 | DOI | Zbl

[2] A. L. Skubachevskii, “O bifurkatsii Khopfa dlya kvazilineinogo parabolicheskogo funktsionalno-differentsialnogo uravneniya”, Differents. uravneniya, 34:10 (1998), 1394–1401 | MR | Zbl

[3] A. L. Skubachevskii, “Bifurcation of periodic solutions for nonlinear parabolic functional differential equations arising in optoelectronics”, Nonlinear Anal., 32:2 (1998), 261–278 | DOI | MR | Zbl

[4] A. L. Skubachevskii, Elliptic Functional Differential Equations and Applications, Birkhäuser, Basel, 1997 | MR | Zbl

[5] A. L. Skubachevskii, “Kraevye zadachi dlya ellipticheskikh funktsionalno-differentsialnykh uravnesnii i ikh prilozheniya”, UMN, 71:5 (2016), 3–112 | DOI | MR | Zbl

[6] A. L. Skubachevskii, “Neklassicheskie kraevye zadachi. I”, SMFN, 26, RUDN, M., 2007, 3–132 | MR | Zbl

[7] A. L. Skubachevskii, “Neklassicheskie kraevye zadachi. II”, Uravneniya v chastnykh proizvodnykh, SMFN, 33, RUDN, M., 2009, 3–179 | MR

[8] P. L. Gurevich, “Ellipticheskie zadachi s nelokalnymi kraevymi usloviyami i polugruppy Fellera”, Uravneniya v chastnykh proizvodnykh, SMFN, 38, RUDN, M., 2010, 3–173 | MR | Zbl

[9] A. B. Muravnik, “Ellipticheskie differentsialno-raznostnye uravneniya v poluprostranstve”, Matem. zametki, 108:5 (2020), 764–770 | DOI | MR | Zbl

[10] A. B. Muravnik, “Ellipticheskie differentsialno-raznostnye uravneniya obschego vida v poluprostranstve”, Matem. zametki, 110:1 (2021), 90–98 | DOI | Zbl

[11] A. B. Muravnik, “Ellipticheskie differentsialno-raznostnye uravneniya s raznonapravlennymi sdvigami v poluprostranstve”, Ufimsk. matem. zhurn., 13:3 (2021), 107–115

[12] A. B. Muravnik, “O zadache Dirikhle v poluploskosti dlya differentsialno-raznostnykh ellipticheskikh uravnenii”, SMFN, 60, RUDN, M., 2016, 102–113

[13] A. B. Muravnik, “Asimptoticheskie svoistva reshenii zadachi Dirikhle v poluploskosti dlya nekotorykh differentsialno-raznostnykh ellipticheskikh uravnenii”, Matem. zametki, 100:4 (2016), 566–576 | DOI | MR | Zbl

[14] A. B. Muravnik, “On the half-plane Dirichlet problem for differential-difference elliptic equations with several nonlocal terms”, Math. Model. Nat. Phenom., 12:6 (2017), 130–143 | DOI | MR | Zbl

[15] A. B. Muravnik, “Asimptoticheskie svoistva reshenii dvumernykh differentsialno-raznostnykh ellipticheskikh zadach”, SMFN, 63, no. 4, Rossiiskii universitet druzhby narodov, M., 2017, 678–688 | DOI

[16] A. B. Muravnik, “Ellipticheskie zadachi s nelokalnym potentsialom, voznikayuschie v modelyakh nelineinoi optiki”, Matem. zametki, 105:5 (2019), 747–762 | DOI | MR | Zbl

[17] A. B. Muravnik, “Half-plane differential-difference elliptic problems with general-kind nonlocal potentials”, Complex Var. Elliptic Equ., 2020 | DOI

[18] I. M. Gelfand, G. E. Shilov, “Preobrazovaniya Fure bystro rastuschikh funktsii i voprosy edinstvennosti resheniya zadachi Koshi”, UMN, 8:6(58) (1953), 3–54 | MR | Zbl

[19] G. E. Shilov, Matematicheskii analiz, Vtoroi spetsialnyi kurs, Izd-vo Mosk. un-ta, M., 1984 | Zbl

[20] I. M. Gelfand, G. E. Shilov, Obobschennye funktsii. Vyp. 3. Nekotorye voprosy teorii differentsialnykh uravnenii, Fizmatgiz, M., 1958 | MR | Zbl