Geodesic
On positive solutions of elliptic equations
Sbornik. Mathematics, Tome 14 (1971) no. 4, pp. 587-613 
T. G. Pletneva; S. D. Èidel'man; V. A. Kondrat'ev. On positive solutions of elliptic equations. Sbornik. Mathematics, Tome 14 (1971) no. 4, pp. 587-613. http://geodesic.mathdoc.fr/item/SM_1971_14_4_a8/
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper the authors study weak solutions of elliptic equations of the form $$ Pu\equiv\sum_{|k|\leqslant m}(-1)^kD_x^k\bigl(a_k(x)u(x)\bigr)=f(x) $$ in a bounded domain $\Omega$. It is assumed known about these solutions either that they are positive, or that estimates in certain norms hold for their negative parts. It is assumed moreover that an estimate on the $~L_1$-norm of the solution holds on some subdomain $\Omega'\subset\Omega$. Summability of such solutions with a weight function that vanishes at the boundary is established, and with the use of the results of Ya. A. Roitberg integral representations are given in terms of the Green's function for the Dirichlet problem. Bibliography: 8 titles.

[1] V. A. Kondratev, S. D. Eidelman, “O svoistvakh reshenii lineinykh evolyutsionnykh sistem s ellipticheskoi prostranstvennoi chastyu”, Matem. sb., 81(123) (1970), 398–429 | MR | Zbl

[2] V. A. Kondratev, S. D. Eidelman, “O nekotorykh svoistvakh polozhitelnykh reshenii evolyutsionnykh gipoellipticheskikh uravnenii”, DAN SSSR, 184:5 (1969), 1027–1030 | MR | Zbl

[3] V. A. Kondratev, S. D. Eidelman, “O kharaktere reshenii lineinykh evolyutsionnykh sistem s ellipticheskoi prostranstvennoi chastyu”, DAN SSSR, 189:3 (1969), 468–471 | MR | Zbl

[4] L. Khermander, Lineinye differentsialnye operatory s chastnymi proizvodnymi, Mir, Moskva, 1965 | MR

[5] Ya. A. Roitberg, “O granichnykh znacheniyakh obobschennykh reshenii ellipticheskikh uravnenii”, DAN SSSR, 188:1 (1969), 41–44 | MR | Zbl

[6] Yu. M. Berezanskii, Razlozhenie po sobstvennym funktsiyam samosopryazhennykh operatorov, Naukova dumka, Kiev, 1965 | MR

[7] Yu. M. Berezanskii, Ya. A. Roitberg, “Teorema o gomeomorfizmakh i funktsiya Grina dlya obschikh ellipticheskikh granichnykh zadach”, Ukr. matem. zh., 19:5 (1967)

[8] O. A. Ladyzhenskaya, V. D. Solonnikov, N. N. Uraltseva, Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, Moskva, 1967