Geodesic
Smoothness of generalized solutions of the Dirichlet problem for strongly elliptic functional differential equations with orthotropic contractions
Contemporary Mathematics. Fundamental Directions, Proceedings of the Seventh International Conference on Differential and Functional-Differential Equations (Moscow, August 22–29, 2014). Part 1, Tome 58 (2015), pp. 153-165 
A. L. Tasevich. Smoothness of generalized solutions of the Dirichlet problem for strongly elliptic functional differential equations with orthotropic contractions. Contemporary Mathematics. Fundamental Directions, Proceedings of the Seventh International Conference on Differential and Functional-Differential Equations (Moscow, August 22–29, 2014). Part 1, Tome 58 (2015), pp. 153-165. http://geodesic.mathdoc.fr/item/CMFD_2015_58_a8/
@article{CMFD_2015_58_a8,
     author = {A. L. Tasevich},
     title = {Smoothness of generalized solutions of the {Dirichlet} problem for strongly elliptic functional differential equations with orthotropic contractions},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {153--165},
     year = {2015},
     volume = {58},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2015_58_a8/}
}
TY  - JOUR
AU  - A. L. Tasevich
TI  - Smoothness of generalized solutions of the Dirichlet problem for strongly elliptic functional differential equations with orthotropic contractions
JO  - Contemporary Mathematics. Fundamental Directions
PY  - 2015
SP  - 153
EP  - 165
VL  - 58
UR  - http://geodesic.mathdoc.fr/item/CMFD_2015_58_a8/
LA  - ru
ID  - CMFD_2015_58_a8
ER  - 
%0 Journal Article
%A A. L. Tasevich
%T Smoothness of generalized solutions of the Dirichlet problem for strongly elliptic functional differential equations with orthotropic contractions
%J Contemporary Mathematics. Fundamental Directions
%D 2015
%P 153-165
%V 58
%U http://geodesic.mathdoc.fr/item/CMFD_2015_58_a8/
%G ru
%F CMFD_2015_58_a8

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

In the disk, we consider the first boundary-value problem for a functional differential equation containing transformations of orthotropic contractions of independent variables of the unknown function. We study the smoothness of generalized solutions inside special-type subdomains and near their boundaries and pose strong ellipticity conditions.

[1] Vishik M. I., “O silno ellipticheskikh sistemakh differentsialnykh uravnenii”, Mat. sb., 29(71):3 (1951), 615–676 | MR | Zbl

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

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

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

[5] Mikhailov V. P., Differentsialnye uravneniya v chastnykh proizvodnykh, Nauka, M., 1976 | MR

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

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

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

[9] 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

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

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

[12] 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

[13] Cby Шамин Р. В. “O prostranstvakh nachalnykh dannykh dlya differentsialnykh uravnenii v gilbertovykh prostranstvakh”, Mat. sb., 194:9 (2003), 141–156 | DOI | MR | Zbl

[14] Auscher C., Hofmann S., McIntosh A., Tchamitchian C., “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

[15] 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

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

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

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

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

[20] Morrey C. B., Multiple Integrals in the Calculus of Variations, Springer, Berlin–Heidelberg–New York, 1966 | MR | Zbl

[21] Neverova D. A., Skubachevskii A. L., “On smoothness of generalized solutions to boundary value problems for strongly elliptic differential-difference equations on a boundary of neighboring domains”, Russ. J. Math. Phys., 22:4 (2015), 504–517 | MR

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

[23] Skubachevskii A. L., Elliptic Functional Differential Equations and Applications, Birkhäuser, Basel–Boston–Berlin, 1997 | MR | Zbl