Расширения пространства непрерывных функций и~их применение к~исследованию задачи Дирихле

A. K. Gushchin

Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 85 (2024) no. 1, pp. 157-181.

Bibliographie

[1] Guschin A. K., “Trudy V. A. Steklova po uravneniyam matematicheskoi fiziki i razvitie ego rezultatov v etoi oblasti”, Tr. MIAN, 289, 2015, 145–162 | DOI

[2] Hopf E., “Elementare Bemerkungen über die Lösungen partieller Differentialgleichungen zweiter Ordnung vom elliptichen Typus”, Sitzungsberichte Preuss Akad. Wissensch., 19, Berlin, 1927, 147–152

[3] Gilbarg D., Trudinger N., Elliptic partial differential equations of second order, Grundlehren math. Wiss. [Fundamental Principles of Mathematical Sciences], 224, Springer-Verlag, Berlin, 1983 | MR

[4] Korn A., Über Minimalflächen, deren Randkurven wenig von ebenen Kurven abwiechen, Abhandl. Preuss. Akad. Wiss. Berlin, 2, 1909

[5] Schauder J., “Über lineare elliptische Differentialgleichungen zweiter Ordnung”, Math. Zeit., 38 (1934), 257–282 | DOI | MR

[6] Schauder J., “Numerische Abschätzungen in elliptischen linearen Differentialgleichungen”, Studia Math., 5 (1935), 34–42 | DOI

[7] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973 | MR

[8] Liapounoff A., “Sur certaines questions qui se rattachent au problème de Dirichlet”, Journal des Mathém. pures et appl. Ser. 5, 4 (1898), 241–311 | MR

[9] Poincaré H., “La méthode de Neumann et le problème de Dirichlet”, Acta Mathematica, 20 (1897), 59–142 | DOI | MR

[10] Korn A., Lehrbuch der Potentialtheorie, v. I, Ferd. Dümmler, Berlin, 1899

[11] Stekloff W., “Sur les problèmes fondamentaux de la physique mathématique”, C. R. Acad. Sci. Paris, 128 (1899), 588–591

[12] Stekloff W. A., “Les méthodes générales pour résoudre les problèmes fondamentaux de la physique mathématique”, AFT. 2 série, 2 (1900), 207–272 | MR

[13] Steklov V. A., Osnovnye zadachi matematicheskoi fiziki, Nauka, M., 1983

[14] Zaremba S., “Sur la théorie de l'équation de Laplace et les méthodes de Neumann et de Robin”, Bull. Acad. Sci. Cracovie, 1901, 171–189

[15] Lebesgue H., “Sur le problème de Dirichlet”, Rend. Circ. Mat. Palermo, 24 (1907), 371–402 | DOI

[16] Perron O., “Eine neue Behandlung der ersten Randwertaufgabe für $\Delta u = 0$”, Math. Zeit., 18 (1923), 42–54 | DOI | MR

[17] Petrovskii I. G., Lektsii ob uravneniyakh s chastnymi proizvodnymi, GITTL, M., 1953 | MR

[18] Wiener N., “The Dirichlet problem”, J. Math. and Phys., 3:3 (1924), 127–146 | DOI | MR

[19] Oleinik O. A., “O zadache Dirikhle dlya uravnenii ellipticheskogo tipa”, Matem. sb., 24:1 (1949), 3–14

[20] Soboleff S., “Méthode nouvelle à résoudre le problème de Cauchy pour les équations linéaires hyperboliques normales”, Matem. sb., 1:1 (1936), 39–72

[21] Sobolev S. L., Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, Izd. 3, Nauka, M., 1988 | MR

[22] Hilbert D., “Über das Dirichlet'sche Princip”, Jber. Deutsch. Math., 8 (1900), 184–188

[23] Kondratev V. A., Landis E. M., “Kachestvennaya teoriya lineinykh differentsialnykh uravnenii v chastnykh proizvodnykh vtorogo poryadka”, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Fundam. napravleniya, 32, VINITI, M., 1988, 99–215

[24] Nechas I., “O resheniyakh ellipticheskikh differentsialnykh uravnenii v chastnykh proizvodnykh vtorogo poryadka s neogranichennym integralom Dirikhle”, Chekhoslov. matem. zhurnal, 10 (1960), 283–298

[25] Riesz F., “Über die Randwerte einer analytischen Funktion”, Math. Z., 18 (1923), 87–95 | DOI | MR

[26] Littlewood J., Paley R., “Theorems on Fourier series and power series. I”, J. Lond. Math. Soc., 6 (1931), 230–233 | DOI | MR

[27] Littlewood J., Paley R., “Theorems on Fourier series and power series. II”, Proc. Lond. Math. Soc., 42 (1937), 52–89 | DOI | MR

[28] Littlewood J., Paley R., “Theorems on Fourier series and power series. III”, Proc. Lond. Math. Soc., 43 (1938), 105–126 | DOI | MR

[29] Privalov I. I., Integral Cauchy, Saratov, 1919

[30] Mikhailov V. P., “O zadache Dirikhle dlya ellipticheskogo uravneniya vtorogo poryadka”, Differents. uravneniya, 12:10 (1976), 1877–1891 | MR

[31] Guschin A. K., Mikhailov V. P., “O suschestvovanii granichnykh znachenii reshenii ellipticheskogo uravneniya”, Matem. sb., 182:6 (1991), 787–810

[32] Guschin A. K., “O zadache Dirikhle dlya ellipticheskogo uravneniya vtorogo poryadka”, Matem. sb., 137:1 (1988), 19–64

[33] Guschin A. K., “Obobscheniya prostranstva nepreryvnykh funktsii; teoremy vlozheniya”, Matem. sb., 211:11 (2020), 54–71 | DOI | MR

[34] Karleson L., Izbrannye problemy teorii isklyuchitelnykh mnozhestv, Mir, M., 1971

[35] Guschin A. K., “Integral ploschadei Luzina i nekasatelnaya maksimalnaya funktsiya dlya reshenii ellipticheskogo uravneniya vtorogo poryadka”, Matem. sb., 209:6 (2018), 47–64 | DOI | MR

[36] Dumanyan V. Zh., “O razreshimosti zadachi Dirikhle dlya obschego ellipticheskogo uravneniya vtorogo poryadka”, Matem. sb., 202:7 (2011), 75–94 | DOI | MR

[37] Dumanyan V. Zh., “O razreshimosti zadachi Dirikhle dlya ellipticheskogo uravneniya vtorogo poryadka”, TMF, 180:2 (2014), 189–205 | DOI | MR

[38] Dumanyan V. Zh., “On solvability of the Dirichlet problem with the boundary function in $L_2$ for a second-order elliptic equation”, J. Contemp. Math. Anal., 50:4 (2015), 153–166 | DOI | MR

[39] Guschin A. K., “O razreshimosti zadachi Dirikhle dlya neodnorodnogo ellipticheskogo uravneniya vtorogo poryadka”, Matem. sb., 206:10 (2015), 71–102 | DOI

[40] Guschin A. K., Mikhailov V. P., “O razreshimosti nelokalnykh zadach dlya ellipticheskogo uravneniya vtorogo poryadka”, Matem. sb., 185:1 (1994), 121–160

[41] Guschin A. K., Mikhailov V. P., “O nepreryvnosti reshenii odnogo klassa nelokalnykh zadach dlya ellipticheskogo uravneniya”, Matem. sb., 186:2 (1995), 37–58 | MR

[42] Guschin A. K., “$L_p$-otsenki resheniya zadachi Dirikhle dlya ellipticheskogo uravneniya vtorogo poryadka”, TMF, 174:2 (2013), 243–255 | DOI | MR

[43] Carleson L., “An interpolation problem for bounded analytic functions”, Amer. J. Math., 80 (1958), 921–930 | DOI | MR

[44] Carleson L., “Interpolations by bounded analytic functions and the corona problem”, Ann. of Math., 76:3 (1962), 547–559 | DOI | MR

[45] Hormander L., “$L^p$ estimates for (pluri-) subharmonic functions”, Math. Scand., 20 (1967), 65–78 | DOI | MR

[46] Giorgi E. de., “Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari”, Mem. Accad. Sci. Torino, 3 (1957), 25–43 | MR

[47] Nash J., “Continuity of solutions of parabolic and elliptic equations”, Amer. J. Math., 80 (1958), 931–954 | DOI | MR

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

[49] Guschin A. K., “O vnutrennei gladkosti reshenii ellipticheskikh uravnenii vtorogo poryadka”, Sib. matem. zhurn., 46:5 (2005), 1036–1052 | MR

[50] Guschin A. K., “Nekotoroe usilenie svoistva vnutrennei nepreryvnosti po Gelderu reshenii zadachi Dirikhle dlya ellipticheskogo uravneniya vtorogo poryadka”, TMF, 157:3 (2008), 345–363 | DOI | MR