A regularity condition for generalized solutions of higher-order quasilinear elliptic equations

I. V. Skrypnik

Izvestiya. Mathematics , Tome 7 (1973) no. 6, pp. 1371-1421.

Résumé

Regularity is proved for an arbitrary generalized solution of a quasilinear elliptic equation of divergent type which belongs to $W_2^{m+n/2}(\Omega')$, for an arbitrary strictly interior subregion $\Omega'$ of a region $\Omega$ ($2m$ is the order of the equation, and $n$ is the number of arguments). It follows from this, in particular, that the regularity problem has an affirmative solution in the two-dimensional case.

Bibliographie

[1] Problemy Gilberta, Nauka, M., 1969 | MR

[2] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1964 | MR

[3] Petrovskii I. G., “Ob analitichnosti reshenii uravnenii s chastnymi proizvodnymi”, Matem. sb., 5:1 (1939), 3–68 | Zbl

[4] Agmon S., Duglis A., Nirenberg L., Otsenki reshenii ellipticheskikh uravnenii vblizi granitsy, IL, M., 1962

[5] Mazya V. G., “Primery neregulyarnykh reshenii kvazilineinykh ellipticheskikh uravnenii s analiticheskimi koeffitsientami”, Funkts. analiz, 2:3 (1968), 53–57 | MR

[6] De Giorgi E., “Un esempio di estremali discontinue per un problema variazionale di tipo elliptico”, Bolletino della Unione Matematica Italiana, ser. IV, 1968, no. 1, 135–137 | MR | Zbl

[7] Giusti E., Miranda M., “Un esempio di soluziono discontinue per un problema di minimo relativo ad un integrate regolare del calcolo delle variazioni”, Bollettino della Unione Matematica Italiana, ser. IV, 1968, no. 2, 219–226 | MR | Zbl

[8] Morrey C. B., “Partial regularity results for non-linear elliptic systems”, J. Math. and Mech., 17:7 (1968), 649–670 | MR | Zbl

[9] Giusti E., Miranda M., “Sulla regolarita delle soluzioni deboli di una class di sistemi ellittici quasi-lineari”, Archive Rat'l. Mech. Anal., 31 (1968), 173–184 | MR | Zbl

[10] Nečas I., “Sur la regularite des solutions faibles des equation elliptiques non lineaires”, Comm. Math. Univ. Carol., 9:3 (1966), 366–113

[11] Fokht A. S., “O differentsialnykh svoistvakh reshenii odnogo klassa kvazilineinykh uravnenii ellipticheskogo tipa”, Uch. zap. Mosk. obl. ped. in-ta, 269, no. 14, 1969, 241–249

[12] Nečas I., “On the demiregularity of weak solutions of non-linear elliptic equations”, Bull. Amer. Math. Soc., 77:1 (1971), 151–156 | DOI | MR | Zbl

[13] Frehse I., “On the boundedneiss of weak solutions of higher order nonlinear elliptic partial differential equations”, Boll. Unione Mat. Italiana, 3:4 (1970), 407–427 | MR

[14] Widmann K., “Hölder continuity of solutions of elliptic systems”, Manuscripta Mathematica, 5:4 (1971), 299–308 | DOI | MR

[15] Nikolskii S. M., Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Nauka, M., 1969 | MR

[16] Skrypnik I. V., “O regulyarnosti obobschennykh reshenii kvazilineinykh ellipticheskikh uravnenii proizvolnogo poryadka”, Dokl. AN SSSR, 203:1 (1972), 36–38. | MR | Zbl

[17] Skrypnik I. V., Kvazilineinye ellipticheskie uravneniya vysshego poryadka, Don. Gos. un-t, 1971

[18] Vishik M. I., “Kvazilineinye silno ellipticheskie sistemy differentsialnykh uravnenii, imeyuschie divergentnuyu formu”, Tr. Mosk. matem. ob-va, 12, 1963, 125–184 | Zbl

[19] Ilin V. P., Solonnikov V. A., “O nekotorykh svoistvakh differentsiruemykh funktsii mnogikh peremennykh”, Tr. matem. in-ta im. V. A. Steklova AN SSSR, 66, 1962, 205–226 | MR

[20] Khardi G. G., Littlvud D. E., Polia G., Neravenstva, IL, M., 1948

[21] Moser I., “A new proof of de Giorgi's theorem concerning the regularity problems for elliptic differential equations”, Comm. Pure and Appl. Math., 13:3 (1960), 457–468 | DOI | MR | Zbl