Regularity of solutions of linear and quasilinear equations of elliptic type in divergence form

F. I. Mamedov

Matematičeskie zametki, Tome 53 (1993) no. 1, pp. 68-82

Résumé

We prove estimates in $C(D)$ and $L_p(D)$ and in Orlicz norms of solutions of the following linear and quasilinear equations: $$ \sum_{i,k=1}^n\frac\partial{\partial x_i}\biggl(a_{ik}(x)\frac{\partial u}{\partial x_k}\biggr)+\sum_{i=1}^n\frac\partial{\partial x_i}(b_i(x)u)+c(x)u=\sum_{i=1}^n\frac{\partial f^i}{\partial x_i} $$ and $$ \sum_{i=1}^n\frac\partial{\partial x_i}\bigl(a_i(x,u,\nabla u)\bigr)+h(x,u)=\sum_{i=1}^n\frac{\partial f^i}{\partial x_i}, $$ depending on the membership of the functions $c(x)$, $b_i(x)$ and $f^i(x)$ in various spaces $L_p(D)$. We write out explicitly the constants in the estimates obtained.