On positive solutions of elliptic equations

T. G. Pletneva ; S. D. Èidel'man ; V. A. Kondrat'ev

Sbornik. Mathematics, Tome 14 (1971) no. 4, pp. 587-613

Résumé

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.