Geodesic
On boundary properties of solutions of elliptic equations in multidimensional domains representable by means of the difference of convex functions
Sbornik. Mathematics, Tome 61 (1988) no. 2, pp. 437-460 Cet article a éte moissonné depuis la source Math-Net.Ru

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The author examines the solution of a linear second order uniformly elliptic equation with variable coefficients defined inside a domain whose boundary is locally representable with the aid of the difference of convex functions (the spatial analog of Radon domain without cusps in the plane). We introduce the concept of "$p$-area integral", generalizing the known Luzin area integral. Local and integral theorems are obtained on the connection between this integral and the nontangential maximal function of the solution, and also the conditions for existence of nontangential boundary values almost everywhere and in the $L_2$-metric. Bibliography: 17 titles. 
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     author = {V. Yu. Shelepov},
     title = {On boundary properties of solutions of elliptic equations in multidimensional domains representable by means of the difference of convex functions},
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     year = {1988},
     volume = {61},
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     url = {http://geodesic.mathdoc.fr/item/SM_1988_61_2_a10/}
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V. Yu. Shelepov. On boundary properties of solutions of elliptic equations in multidimensional domains representable by means of the difference of convex functions. Sbornik. Mathematics, Tome 61 (1988) no. 2, pp. 437-460. http://geodesic.mathdoc.fr/item/SM_1988_61_2_a10/

[1] Burkholder D. L., Gundy R. F., “Distribution function inequalities for the area integral”, Studia Math., 44:6 (1972), 527–544 | MR | Zbl

[2] Mikhailov V. P., “O granichnykh znacheniyakh reshenii ellipticheskikh uravnenii v oblastyakh s gladkoi granitsei”, Matem. sb., 101(143) (1976), 163–188

[3] Petrushko I. M., “O granichnykh znacheniyakh reshenii ellipticheskikh uravnenii v oblastyakh s lyapunovskoi granitsei”, Matem. sb., 119(161) (1982), 48–77 | MR | Zbl

[4] Shelepov V. Yu., “Ob integrale Luzina i netangentsialnoi maksimalnoi funktsii v oblastyakh, granitsy kotorykh predstavimy s pomoschyu raznosti vypuklykh funktsii”, DAN SSSR, 283 (1985), 822–825 | MR | Zbl

[5] Dahlberg B. E. J., “Weighted norm inequalities for the Lusin area integral and the nontangential maximal functions for functions harmonic in Lipschits domain”, Studia Math., 67:3 (1980), 297–314 | MR | Zbl

[6] Mazya V. G., “O vyrozhdayuscheisya zadache s kosoi proizvodnoi”, Matem. sb., 87(129) (1972), 417–454

[7] Gorbachuk V. I., Gorbachuk M. L., Granichnye zadachi dlya differentsialno-operatornykh uravnenii, Naukova dumka, Kiev, 1984 | MR | Zbl

[8] Roitberg Ya. A., “O suschestvovanii predelnykh znachenii obobschennykh reshenii ellipticheskikh uravnenii na granitse oblasti”, Sib. matem. zhurn., 20:2 (1979), 386–396 | MR | Zbl

[9] Guschin A. K., Mikhailov V. P., “O granichnykh znacheniyakh v $L_p$, $p>1$, reshenii ellipticheskikh uravnenii”, Matem. sb., 108(150) (1979), 3–21 | MR | Zbl

[10] Stein I. M., Singulyarnye integraly i differentsialnye svoistva funktsii, Mir, M., 1973 | MR

[11] Koshelev A. I., “Ob ogranichennosti v $L_p$ proizvodnykh reshenii ellipticheskikh differentsialnykh uravnenii”, Matem. sb., 38(80) (1956), 359–372 | Zbl

[12] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1978 | MR

[13] Bakelman I. Ya., Verner A. A., Kantor B. E., Vvedenie v differentsialnuyu geometriyu v tselom, Nauka, M., 1973 | MR

[14] Calderon A. P., “On a theorem of Marcinkiewicz and Zygmund”, Trans. Amer. Math. Soc., 68:1 (1950), 55–61 | DOI | MR | Zbl

[15] Danilyuk I. I., Neregulyarnye granichnye zadachi, Nauka, M., 1975 | MR

[16] Aleksandrov A. D., “O poverkhnostyakh, predstavimykh raznostyu vypuklykh funktsii”, Izv. AN KazSSR. Ser. matem. i mekh., 60:3 (1949), 3–20

[17] Krasnoselskii M. A., Rutitskii L. B., Vypuklye funktsii i prostranstva Orlicha, Fizmatgiz, M., 1958 | MR