Geodesic
On exceptional sets on the boundary and the uniqueness of solutions of the Dirichlet problem for a second order elliptic equation
Sbornik. Mathematics, Tome 39 (1981) no. 1, pp. 107-123 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The Dirichlet problem is considered for a linear elliptic equation of second order in $n$-dimensional domain $Q$, $n\geqslant2$, with smooth boundary $\partial Q$ in the case where the generalized solution of this equation takes boundary values everywhere on the boundary but an exceptional set $\mathscr E\subset\partial Q$. It is proved that for $n/(n-1)\leqslant p<\infty$ the space $L_p(Q)$ is a class of uniqueness for such a problem if $\mathscr E$ has finite Hausdorff measure of order $n-q$, where $\frac1p+\frac1q=1$. By an example of the Dirichlet problem for Laplace's equation it is shown that the indicated order of the Hausdorff measure is best possible. Bibliography: 14 titles. 
@article{SM_1981_39_1_a4,
     author = {S. V. Gaidenko},
     title = {On exceptional sets on the boundary and the uniqueness of solutions of the {Dirichlet} problem for a~second order elliptic equation},
     journal = {Sbornik. Mathematics},
     pages = {107--123},
     year = {1981},
     volume = {39},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1981_39_1_a4/}
}
TY  - JOUR
AU  - S. V. Gaidenko
TI  - On exceptional sets on the boundary and the uniqueness of solutions of the Dirichlet problem for a second order elliptic equation
JO  - Sbornik. Mathematics
PY  - 1981
SP  - 107
EP  - 123
VL  - 39
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1981_39_1_a4/
LA  - en
ID  - SM_1981_39_1_a4
ER  - 
%0 Journal Article
%A S. V. Gaidenko
%T On exceptional sets on the boundary and the uniqueness of solutions of the Dirichlet problem for a second order elliptic equation
%J Sbornik. Mathematics
%D 1981
%P 107-123
%V 39
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1981_39_1_a4/
%G en
%F SM_1981_39_1_a4
S. V. Gaidenko. On exceptional sets on the boundary and the uniqueness of solutions of the Dirichlet problem for a second order elliptic equation. Sbornik. Mathematics, Tome 39 (1981) no. 1, pp. 107-123. http://geodesic.mathdoc.fr/item/SM_1981_39_1_a4/

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

[2] V. P. Mikhailov, “O granichnykh znacheniyakh reshenii ellipticheskogo uravneniya v share”, Matem. sb., 100(142) (1976), 5–13

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

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

[5] B. E. J. Dahlberg, “On exeptional sets at the boundary for subharmonic functions”, Arkiv mat., 15:2 (1977), 305–312 | DOI | MR | Zbl

[6] L. Karleson, Izbrannye problemy teorii isklyuchitelnykh mnozhestv, izd-vo Mir, Moskva, 1971 | MR

[7] R. Harvey, J. Polking, “Removable singularities of solutions of linear partial differential equations”, Acta Math., 125 (1970), 39–56 | DOI | MR | Zbl

[8] V. G. Mazya, V. P. Khavin, “Prilozheniya $(p,l)$-emkosti k neskolkim zadacham teorii isklyuchitelnykh mnozhestv”, Matem. sb., 90(132) (1973), 558–591

[9] L. Carleson, “Sets of uniqueness fir functions regular in the unite circle”, Acta Math., 87 (1952), 325–345 | DOI | MR | Zbl

[10] E. I. Moiseev, “O suschestvennykh i nesuschestvennykh granichnykh mnozhestvakh zadachi Neimana”, Diff. uravneniya, 9:5 (1973), 901–911 | MR | Zbl

[11] A. I. Koshelev, “Apriornye otsenki v $L_p$ i obobschennye resheniya ellipticheskikh uravnenii i sistem”, Uspekhi matem. nauk, XIII:4(82) (1958), 29–88 | MR | Zbl

[12] F. E. Browder, “Estimates and existence theorems for elliptic boundary value problems”, Proc. Nat. Acad. Sci. USA, 45:3 (1959), 365–372 | DOI | MR | Zbl

[13] E. Kollingvud, A. Lovater, Teoriya predelnykh mnozhestv, izd-vo “Mir”, Moskva, 1971 | MR

[14] V. P. Mikhailov, Differentsialnye uravneniya v chastnykh proizvodnykh, izd-vo “Nauka”, Moskva, 1976 | MR