On comparison theorems for quasi-linear elliptic inequalities with a~special account of the geometry of the domain

A. A. Kon'kov

Izvestiya. Mathematics , Tome 78 (2014) no. 4, pp. 758-808

Résumé

We consider non-negative solutions of quasi-linear elliptic inequalities $\operatorname{div}A(x,Du)\geqslant0$ in $\Omega_{R_0,R_1}$, $0\leqslant R_0$, where $\Omega_{R_0,R_1}=\{x\in\Omega\colon R_0|x|$, $\Omega\subset{\mathbb R}^n$ ($n\geqslant2$) is a non-empty open set, and the function $A\colon\Omega_{R_0,R_1}\times{\mathbb R}^n\to{\mathbb R}^n$ satisfies the ellipticity conditions $C_1|\xi|^p\le\xi A(x,\xi)$, $|A(x,\xi)|\le C_2|\xi|^{p-1}$, $C_1,C_2>0$, $p>1$, for almost all $x\in\Omega_{R_0,R_1}$ and all $\xi\in{\mathbb R}^n$. Our bounds for solutions take the geometry of $\Omega$ into account.

Détail

Keywords: non-linear elliptic operators, unbounded domains, capacity.