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

摘要

We consider non-negative solutions of quasi-linear elliptic inequalities divA(x,Du) ≥ 0 in ??_R_0, _R_1, 0 ≤ R_0 < R_1 ≤ ∞, where ?_R_0,_R_1= {x∈?: R_0 < |x| < R_1}, ? ? ?~n (n ≥ 2) is a non-empty open set, and the function A:?_R_0, _R_1 × ?~n → ?~n satisfies the ellipticity conditions C_1|ξ|~p ≤ ξA(x, ξ), |A(x, ξ)| ≤ C_2|ξ|~(p-1), C_1,C_2 > 0, p > 1, for almost all x ∈?: _R_0, _R_1, and all ξ 2 ?~n. Our bounds for solutions take the geometry of ? into account.