On the existence of boundary values of solutions of an elliptic equation

A. K. Gushchin   ; V. P. Mikhailov

Sbornik. Mathematics, Tome 73 (1992) no. 1, pp. 171-194

Résumé

A test is established for the existence of a boundary value for the solution of the elliptic second order equation $$ -\sum_{i,j=1}^n(a_{ij}(x)u_{x_i}(x))_{x_j}=f(x)-\operatorname{div}F(x), \quad x\in Q. $$ In this connection, it is proved that the solution has a property $(u\in C_{n-1}(\overline Q))$ similar to continuity with respect to all variables in $\overline Q$, and that its boundary value $u|_{\partial Q}\in L_2(\partial Q)$ is the limit in $L_2$ of the traces of the solution on surfaces in a large class (which are not necessarily “parallel” to the boundary).

Bibliographie

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

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

[3] Littlewood J., Paley R., “Theorems on Fourier series and power series, II”, J. Lond. Math. Soc., 42 (1936), 52–89 | MR | Zbl

[4] Littlewood J., Paley R., “Theorems on Fourier series and power series, III”, J. Lond. Math. Soc., 43 (1937), 105–126 | MR | Zbl

[5] Privalov I. I., Granichnye svoistva analiticheskikh funktsii, Gostekhizdat, M., 1950

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

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

[8] Mikhailov V. P., “O povedenii vblizi granitsy reshenii ellipticheskogo uravneniya”, DAN SSSR, 226:6 (1976), 1264–1266 | MR

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

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

[11] Mikhailov V. P., “O granichnykh svoistvakh reshenii ellipticheskikh uravnenii”, Matem. zametki, 27:1 (1980), 137–145 | MR

[12] Mihailov V. P., “On the behaviour near boundary of the solution of elliptic equation”, Appl. Math. and Optimiz., 6 (1980), 189–197 | DOI

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

[14] 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

[15] 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

[16] Roitberg Ya. A., “O predelnykh znacheniyakh po poverkhnostyam, parallelnym granitse, obobschennykh reshenii ellipticheskikh uravnenii”, DAN SSSR, 238:6 (1978), 1303–1306 | MR | Zbl

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

[18] 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

[19] Guschin A. K., “O zadache Dirikhle dlya ellipticheskogo uravneniya vtorogo poryadka”, Matem. sb., 137 (179) (1988), 19–64

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