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On Dirichlet problem
Teoretičeskaâ i matematičeskaâ fizika, Tome 218 (2024) no. 1, pp. 60-79 Cet article a éte moissonné depuis la source Math-Net.Ru

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During almost two centuries after the Gauss' formulation of the Dirichlet problem for Laplace equation, many famous mathematicians devoted their studies to this subject and to its various generalizations. Many interesting and important results have been obtained, which become already classical ones. Our paper is an extended presentation of the author's talk on the international conference dedicate to the century of V. S. Vladimirov birthday. Its main content is the review of the results in that direction, including the proves of new statements and discussion of unsolved problems. Our goal is to convince readers that, in this “principal” problem of mathematical physics, we know far from everything even about the case of linear equation. There are many interesting and important unsolved problems in that direction.
Mots-clés : elliptic equation
Keywords: Dirichlet problem, boundary value, Carleson measures. 
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A. K. Gushchin. On Dirichlet problem. Teoretičeskaâ i matematičeskaâ fizika, Tome 218 (2024) no. 1, pp. 60-79. http://geodesic.mathdoc.fr/item/TMF_2024_218_1_a3/

[1] A. K. Guschin, “Trudy V. A. Steklova po uravneniyam matematicheskoi fiziki i razvitie ego rezultatov v etoi oblasti”, Izbrannye voprosy matematiki i mekhaniki, Sbornik statei. K 150-letiyu so dnya rozhdeniya akademika Vladimira Andreevicha Steklova, Trudy MIAN, 289, MAIK “Nauka/Interperiodika”, M., 2015, 145–162 | DOI | DOI

[2] A. M. Lyapunov, “O nekotorykh voprosakh, svyazannykh s zadachei Dirikhle”, Izbrannye trudy, L., 1948, 97–178 | Zbl

[3] H. Poincaré, “La méthode de Neumann et le probléme de Dirichlet”, Acta Math., 20:1 (1896), 59–142 | DOI | MR

[4] A. Korn, Lehrbuch der Potentialtheorie. Band I. Allgemeine Theorie des Potentials und der Potentialfunctionen im Raume, Ferd. Dümmlers Verlagsbuchhanblung, Berlin, 1899 | Zbl

[5] S. Zaremba, “Sur la théorie de l'équation de Laplace et les méthodes de Neumann et de Robin”, Bull. Acad. Sci. Cracovie, 41 (1901), 171–189

[6] V. A. Steklov, Osnovnye zadachi matematicheskoi fiziki, Pod redaktsiei V. S. Vladimirova, Nauka, M., 1983 | MR

[7] M. H. Lebesque, “Sur le problème de Dirichlet”, Rend. Circ. Matem. Palermo, 24 (1907), 371–402 | DOI

[8] O. Perron, “Eine neue Behandlung der Randwertaufgebe für $\Delta u = 0$”, Math. Z., 18:1 (1923), 42–54 | DOI | MR

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

[10] N. Wiener, “The Dirichlet problem”, J. Math. Phys., 3 (1924), 127–146 | DOI | Zbl

[11] O. A. Oleinik, “O zadache Dirikhle dlya uravnenii ellipticheskogo tipa”, Matem. sb., 24(66):1 (1949), 3–14 | MR | Zbl

[12] M. V. Keldysh, “O razreshimosti i ustoichivosti zadachi Dirikhle”, UMN, 1941, no. 8, 171–231 | MR | Zbl

[13] A. Korn, “Zwei Anwendungen der Methode der sukzessiven Annäherungen”, Mathematische Abhandlungen Hermann Amandus Schwarz, eds. C. Caratheodory, G. Hessenberg, E. Landau, L. Lichtenstein, Springer, Berlin, Heidelberg, 1914, 215–229 | DOI

[14] G. Giraud, “Sur le problème de Dirichlet généralisé (deuxième mémoire)”, Ann. Sci. École Norm. Sup. Ser. 3, 46 (1929), 131–245 | DOI | MR

[15] G. Giraud, “Sur certains problèmes non linéaires de Neumann et sur certains problèmes non linéaires mixtes”, Ann. Sci. École Norm. Sup. Ser. 3, 49 (1932), 1–104 | DOI | MR | Zbl

[16] G. Giraud, “Sur certains problèmes non linéaires de Neumann et sur certains non linéaires mixtes”, Ann. Sci. École Norm. Sup. Sup. Ser. 3, 49 (1932), 245–309 | DOI | MR

[17] J. Schauder, “Über lineare elliptische Differentialgleichungen zweiter Ordnung”, Math. Z., 38:1 (1934), 257–282 | DOI | MR

[18] J. Schauder, “Numerische Abschätzungen in elliptischen linearen Differentialgleichungen”, Studia Math., 5 (1934), 34–42 | DOI

[19] A. Korn, Über Minimalflächen, deren Randkurven wenig von ebenen Kurven abwiechen, Königl. Akad. Wiss., Berlin | Zbl

[20] E. Hopf, “Über den funkitonalen, insbesondere den analytischen Charakter der Lösungen elliptichen Differentialgleichungen zweiter Ordnung”, Math. Z., 34:1 (1932), 194–233 | DOI | MR

[21] D. Gilbarg, N. Trudinger, Ellipticheskie differentsialnye uravneniya s chastnymi proizvodnymi vtorogo poryadka, Nauka, M., 1989 | MR

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

[23] D. Hilbert, “Über das Dirichletsche Prinzip”, Jahresber. Deutsch. Math.-Ver., 8:1 (1900), 184–188 | Zbl

[24] S. L. Sobolev, “O nekotorykh otsenkakh, otnosyaschikhsya k semeistvam funktsii, imeyuschikh proizvodnye, integriruemye s kvadratom”, Dokl. AN SSSR, 1(10):7(84) (1936), 267–270

[25] S. Soboleff, “Méthode nouvelle à résoudre le problème de Cauchy pour les équations linéaires hyperboliques normales lineaires hyperboliques normales ??”, Matem. sb., 1(43):1 (1936), 39–72 | Zbl

[26] S. L. Sobolev, “Ob odnoi teoreme funktsionalnogo analiza”, Matem. sb., 4(46):3 (1938), 471–497 | Zbl

[27] S. L. Sobolev, Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, Nauka, M., 1988 | DOI | MR | MR

[28] V. G. Mazya, Prostranstva S. L. Soboleva, Izd-vo Leningr. un-ta, L., 1985 | MR

[29] V. P. Mikhailov, Differentsialnye uravneniya v chastnykh proizvodnykh, Nauka, M., 1983 | MR | MR | Zbl

[30] E. De Giorgi, “Sulla differenziabilita e l'analiticità delle estremali degli integrali multipli regolari”, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3), 3 (1957), 25–43 | MR

[31] J. Nash, “Continuity of solutions of parabolic and elliptic equations”, Amer. J. Math., 80:4 (1958), 931–954 | DOI | MR

[32] J. Moser, “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

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

[34] Yu. A. Alkhutov, V. A. Kondratev, “Razreshimost zadachi Dirikhle dlya ellipticheskikh uravnenii vtorogo poryadka v vypukloi oblasti”, Differ. uravneniya, 28:5 (1992), 806–818 | MR

[35] Yu. A. Alkhutov, “$L_p$-otsenki resheniya zadachi Dirikhle dlya ellipticheskikh uravnenii vtorogo poryadka”, Matem. sb., 189:1 (1998), 3–20 | DOI | DOI | MR | Zbl

[36] V. A. Kondratev, E. M. Landis, “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 | MR | Zbl

[37] I. Nechas, “O resheniyakh ellipticheskikh differentsialnykh uravnenii v chastnykh proizvodnykh vtorogo poryadka s neogranichennym integralom Dirikhle”, Chekhoslov. matem. zhurn., 10(85):2 (1960), 283–298 | DOI

[38] F. Riesz, “Über die Randwerte einer analytischen Funktion”, Math. Z., 18:1 (1923), 87–95 | DOI | MR

[39] J. E. Littlewood, R. E. A. C. Paley, “Theorems on Fourier series and power series”, J. London Math. Soc., s1-6:3 (1931), 230–233 | DOI | MR

[40] J. E. Littlewood, R. E. A. C. Paley, “Theorems on Fourier series and power series (II)”, J. London Math. Soc., s2-42:1 (1937), 52–89 | DOI | MR

[41] J. E. Littlewood, R. E. A. C. Paley, “Theorems on Fourier series and power series (III)”, J. London Math. Soc., s2-43:1 (1938), 105–126 | DOI | MR

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

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

[44] A. K. Guschin, “O zadache Dirikhle dlya ellipticheskogo uravneniya vtorogo poryadka”, Matem. sb., 137(179):1(9), 19–64 | DOI | MR | Zbl

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

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

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

[48] V. Zh. Dumanyan, “O razreshimosti zadachi Dirikhle s granichnoi funktsiei iz $L_2$ dlya ellipticheskogo uravneniya vtorogo poryadka”, Izv. NAN Armenii. Matem., 50:4 (2015), 3–22 | DOI | MR

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

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

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

[52] L. Carleson, “An interpolation problem for analytic functions”, Amer. J. Math., 80:4 (1958), 921–930 | DOI | MR

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

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