Smoothness of generalized solutions to the Dirichlet problem for strongly elliptic functional differential equations with orthotropic contractions on the boundary of adjacent subdomains

A. L. Tasevich

Contemporary Mathematics. Fundamental Directions, Contemporary Mathematics. Fundamental Directions, Tome 69 (2023) no. 1, pp. 152-165.

Résumé

The paper is devoted to the study of the smoothness of generalized solutions of the first boundary-value problem for a strongly elliptic functional differential equation containing orthotropic contraction transformations of the arguments of the unknown function in the leading part. The problem is considered in a circle, the coefficients of the equation are constant. Orthotropic contraction is understood as different contraction in different variables. Conditions for the conservation of smoothness on the boundaries of neighboring subdomains formed by the action of the contraction transformation group on a circle are found in explicit form for any right-hand side from the Lebesgue space.

Détail

Keywords: strongly elliptic functional differential equation, orthotropic contraction of arguments, smoothness of generalized solutions.

Bibliographie

[1] Vishik M. I., “O silno ellipticheskikh sistemakh differentsialnykh uravnenii”, Mat. sb., 29:3 (1951), 615–676

[2] Guseva O. V., “O kraevykh zadachakh dlya silno ellipticheskikh sistem”, Dokl. AN SSSR, 102:6 (1955), 1069–1072 | Zbl

[3] Danford N., Shvarts Dzh. T., Lineinye operatory, v. 2, Mir, M., 1966

[4] Ivanov N. O., Skubachevskii A. L., “Vtoraya kraevaya zadacha dlya differentsialno-raznostnykh uravnenii”, Dokl. RAN, 500 (2021), 74–77 | DOI | Zbl

[5] Ivanov N. O., Skubachevskii A. L., “Ob obobschennykh resheniyakh vtoroi kraevoi zadachi dlya differentsialno-raznostnykh uravnenii s peremennymi koeffitsientami”, Sovrem. mat. Fundam. napravl., 67:3 (2021), 576–595 | MR

[6] Ivanov N. O., Skubachevskii A. L., “Ob obobschennykh resheniyakh vtoroi kraevoi zadachi dlya differentsialno-raznostnykh uravnenii s peremennymi koeffitsientami na intervale netseloi dliny”, Mat. zametki, 111:6 (2022), 873–886 | DOI | Zbl

[7] Ladyzhenskaya O. A., Kraevye zadachi matematicheskoi fiziki, Nauka, M., 1973 | MR

[8] Mikhailov V. P., Differentsialnye uravneniya v chastnykh proizvodnykh, Nauka, M., 1976

[9] Neverova D. A., “Gladkost obobschennykh reshenii vtoroi i tretei kraevykh zadach dlya silno ellipticheskikh differentsialno-raznostnykh uravnenii”, Sovrem. mat. Fundam. napravl., 65, no. 4, 2019, 655–671

[10] Neverova D. A., “Gladkost obobschennykh reshenii zadachi Neimana dlya silno ellipticheskogo differentsialno-raznostnogo uravneniya na granitse sosednikh podoblastei”, Sovrem. mat. Fundam. napravl., 66, no. 2, 2020, 272–291

[11] Rossovskii L. E., “Koertsitivnost funktsionalno-differentsialnykh uravnenii”, Mat. zametki, 59:1 (1996), 103–113 | DOI | MR | Zbl

[12] Rossovskii L. E., “K voprosu o koertsitivnosti funktsionalno-differentsialnykh uravnenii”, Sovrem. mat. Fundam. napravl., 45, 2012, 122–131

[13] Rossovskii L. E., “Ellipticheskie funktsionalno-differentsialnye uravneniya so szhatiem i rastyazheniem argumertov neizvestnoi funktsii”, Sovrem. mat. Fundam. napravl., 54, 2014, 3–138

[14] Rossovskii L. E., Tasevich A. L., “Pervaya kraevaya zadacha dlya silno ellipticheskogo funktsionalno-differentsialnogo uravneniya s ortotropnymi szhatiyami”, Mat. zametki, 97:5 (2015), 733–748 | DOI | MR | Zbl

[15] Skubachevskii A. L., “Gladkost obobschennykh reshenii pervoi kraevoi zadachi dlya ellipticheskogo differentsialno-raznostnogo uravneniya”, Mat. zametki, 34:1 (1983), 105–112 | MR

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

[17] Skubachevskii A. L., Tsvetkov E. L., “Vtoraya kraevaya zadacha dlya ellipticheskikh differentsialno-raznostnykh uravnenii”, Diff. uravn., 25:10 (1989), 1766–1776 | MR | Zbl

[18] Tasevich A. L., “Gladkost obobschennykh reshenii zadachi Dirikhle dlya silno ellipticheskikh funktsionalno-differentsialnykh uravnenii s ortotropnymi szhatiyami”, Sovrem. mat. Fundam. napravl., 58, 2015, 153–165

[19] Tsvetkov E. L., “O gladkosti obobschennykh reshenii tretei kraevoi zadachi dlya ellipticheskogo differentsialno-raznostnogo uravneniya”, Ukr. mat. zh., 45:8 (1993), 1140–1150 | MR | Zbl

[20] Shamin R. V., “O prostranstvakh nachalnykh dannykh dlya differentsialnykh uravnenii v gilbertovykh prostranstvakh”, Mat. sb., 194:9 (2003), 141–156 | DOI | MR | Zbl

[21] Auscher P., Hofmann S., McIntosh A., Tchamitchian P., “The Kato square root problem for higher order elliptic operators and systems on $\mathbb{R}^n$”, J. Evol. Equ., 1:4 (2001), 361–385 | DOI | MR | Zbl

[22] Axelsson A., Keith S., McIntosh A., “The Kato square root problem for mixed boundary value problems”, J. Lond. Math. Soc., 74 (2006), 113–130 | DOI | MR | Zbl

[23] Gårding L., “Dirichlet's problem for linear elliptic partial differential equations”, Math. Scand., 1 (1953), 55–72 | DOI | MR

[24] Kato T., “Fractional powers of dissipative operators”, J. Math. Soc. Japan, 13:3 (1961), 246–274 | DOI | MR | Zbl

[25] Lions J. L., “Espaces d'interpolation et domaines de puissance fractionnaires d'operateurs”, J. Math. Soc. Japan, 14:2 (1962), 233–241 | DOI | MR | Zbl

[26] McIntosh A., “On the comparability of $A^{1/2}$ and $A*^{1/2}$”, Proc. Am. Math. Soc., 32:2 (1972), 430–434 | MR

[27] Morrey C. B., Multiple integrals in the calculus of variations, Springer, Berlin—Heidelberg—New York, 1966 | MR | Zbl

[28] Skubachevskii A. L., “The first boundary value problem for strongly elliptic differential-difference equations”, J. Differ. Equ, 63 (1986), 332–361 | DOI | MR | Zbl

[29] Skubachevskii A. L., Elliptic functional differential equations and applications, Birkhäuser, Basel—Boston—Berlin, 1997 | MR | Zbl