On the Dirichlet problem for a second-order elliptic equation

A. K. Gushchin

Sbornik. Mathematics, Tome 65 (1990) no. 1, pp. 19-66

Résumé

The function space $C_{n-1}(\overline Q)$, $C(\overline Q)\subset C_{n-1}(\overline Q)\subset L_2(Q)$, where $Q$ is a bounded domain in $\mathbf R_n$, consists of elements that on sets of positive $(n-1)$-dimensional Hausdorff measure have traces with a property analogous to joint continuity. For $\partial Q\in C^1$ the set of traces of the functions in $C_{n-1}(\overline Q)$ on $\partial Q$ coincides with $L_2(\partial Q)$, and the imbedding $W_2^1(Q)\subset C_{n-1}(\overline Q)$ is valid. Solutions of the Dirichlet problem in $C_{n-1}(\overline Q)$ are considered for the elliptic equation $$ \sum_{i,j=1}^n(a_{ij}(x)u_{x_i})_{x_j}=f,\quad x\in Q;\qquad u|_{\partial Q}=u_0. $$ Under the assumption that the normal to $\partial Q$ and the coefficients of the equation satisfy the Dini condition on $\partial Q$, it is established that for all $u_0\in L_2(\partial Q)$ and $f\in W_2^{-1}(Q)$ there is a unique solution that depends continuously on $u_0$ and $f$. It is proved that in this situation the solution in $C_{n-1}(\overline Q)$ coincides with the concept of a solution in $W^1_{2,\mathrm{loc}}$ introduced by Mikhailov. Bibliography: 39 titles.

Bibliographie

[1] Sobolev S. L., “Ob odnoi kraevoi zadache dlya poligarmonicheskikh uravnenii”, Matem. sb., 2(44):3 (1937), 465–499 | MR | Zbl

[2] Sobolev S. L., Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, Izd-vo LGU, L., 1950

[3] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1967 | MR

[4] Landis E. M., Uravneniya vtorogo poryadka ellipticheskogo i parabolicheskogo tipov, Nauka, M., 1971 | MR

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

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

[7] Guschin A. K., Mikhailov V. P., “O granichnykh znacheniyakh reshenii ellipticheskikh uravnenii. Obobschennye funktsii i ikh primeneniya v matematicheskoi fizike”, Tr. Mezhdunar. konferentsii, VTs AN SSSR, M., 1981, 189–205

[8] Bogoyavlenskii O. I., Vladimirov V. S, Volovich I. V., Guschin A. K., Drozhzhinov Yu. N., Zharinov V. V., Mikhailov V. P., “Kraevye zadachi matematicheskoi fiziki”, Tr. MIAN., 175 (1986), 63–102 | MR | Zbl

[9] Cimmino G., “Nuovo tipo di condizione al contorno e nuovo metodo perill problema generalizzato di Dirichlet”, Rend. Circ. mat. Palermo, 61 (1938), 177–221 | DOI | Zbl

[10] Miranda K., Uravneniya s chastnymi proizvodnymi ellipticheskogo tipa, IL, M., 1957

[11] Nechas I., “O resheniyakh ellipticheskikh uravnenii vtorogo poryadka s neogranichennym integralom Dirikhle”, Chekhosl. matem. zhurn., 10 (1960), 283–298 | Zbl

[12] Gaidenko S. V., “Ob isklyuchitelnykh mnozhestvakh na granitse i edinstvennosti reshenii zadachi Dirikhle dlya ellipticheskikh uravnenii vtorogo poryadka”, Matem. sb., 111(153) (1980), 116–134 | MR | Zbl

[13] Seeley R., “Singular integrals and boundary value problems”, Amer. J. Math., 88:4 (1966), 781–809 | DOI | MR | Zbl

[14] Seeley R., “Topics in pseudo-differential operators Pseudo-Differential Operators”, Edizioni Gremonese (C. I. M. E., Stresa, 1968), Rome, 1969, 167–305 | MR

[15] Guschin A. K., Mikhailov V. P., “O granichnykh znacheniyakh reshenii ellipticheskikh uravnenii”, Matem. sb., 108(150) (1979), 3–21 | Zbl

[16] Mikhailov Yu. A., “O granichnykh znacheniyakh v $L_p$, $p>1$, reshenii lineinogo ellipticheskogo uravneniya vtorogo poryadka”, Differents. uravneniya, 19:2 (1983), 318–337 | MR

[17] Kondratev V. A., Kopachek I., Oleinik O. A., “O nailuchshikh pokazatelyakh Gëldera dlya obobschennykh reshenii zadachi Dirikhle dlya ellipticheskogo uravneniya vtorogo poryadka”, Matem. sb., 131(173) (1986), 113–125 | MR

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

[19] Petrushko I. M., “O granichnykh znacheniyakh v $L_p$, $p>1$, reshenii ellipticheskikh uravnenii v oblastyakh s lyapunovskoi granitsei”, Matem. sb., 120(162) (1983), 569–588 | MR | Zbl

[20] Karleson L., Izbrannye problemy teorii isklyuchitelnykh mnozhestv, Mir, M., 1971 | MR | Zbl

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

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

[23] Hörmander L., “$L^p$-estimates for (pluri-) subharmonic functions”, Math. scand., 20 (1967), 65–78 | MR | Zbl

[24] Nikolskii N. K., Lektsii ob operatore sdviga, Nauka, M., 1980 | MR

[25] Garnet Dzh., Ogranichennye analiticheskie funktsii, Mir, M., 1984 | MR | Zbl

[26] Mazya V. G., Prostranstva S. L. Soboleva, Izd-vo LGU, L., 1985 | MR

[27] Federer H., Geometric measure theory, Berlin, Heidelberg, New-York, 1969 | MR

[28] Stein E. M., “On the functions of Littlwood–Paley, Lusin and Marcinkiewicz”, Trans. Amer. Math. Soc., 88 (1958), 430–466 | DOI | MR

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

[30] Fefferman C., Stein E. M., “$H^p$ spaces of several variables”, Acta math., 129 (1972), 137–192 | DOI | MR

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

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

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

[34] Shelepov V. Yu., “O granichnykh svoistvakh reshenii ellipticheskikh uravnenii v mnogomernykh oblastyakh, predstavlyaemykh s pomoschyu raznosti vypuklykh funktsii”, Matem. sb., 133(175) (1987), 446–468 | Zbl

[35] Guschin A. K., “O zadache Dirikhle dlya ellipticheskogo uravneniya. Nelineinye zadachi matematicheskoi fiziki”, Tezisy dokladov VI Respublikanskoi konferentsii, IPMM AN USSR, Donetsk, 1987, 39

[36] Frostman O., “Potential d'equilibre et capacité des ensembles avec quelques applications à la théorie des fonctions”, Meddel Lunds Unov. Mat. Sem., 3 (1935), 1–118

[37] Federer H., “The area of non-parametric surfaces”, Proc. Amer. Math. Soc., 11:3 (1960), 436–439 | DOI | MR | Zbl

[38] Hardy G. H., Littlewood J. E., “A maximal theorem with function-theoretic applications”, Acta Math., 54 (1930), 81–116 | DOI | MR | Zbl

[39] Skubachevskii A. L., “Ellipticheskie zadachi s nelokalnymi usloviyami vblizi granitsy”, Matem. sb., 129(171) (1986), 279–302 | MR