Non-homogeneous divergence structure inequalities

Abstract

We consider the quasilinear differential inequality divA(x,u,Du) + B(x,u,Du) ≥ 0 in Ω,(6.1.1) where Ω is a bounded domain in ℝn, and A and B satisfy the generic assumptions of Section 3.1. Here we shall extend the validity of Theorems 3.2.1 and 3.2.2 to the case when (6.1.1) is inhomogeneous, that is, there are constants a2, b1, b2, a, b ≥ 0 such that for all (x, z, ξ) ∈ Ω × ℝ+ × ℝn there holds, for p > 1,(Formula Presented) (6.1.2) while for p = 1, (Formula Presented) (6.1.3)(in (6.1.3) we write b for b2 and discard the terms b1|ξ|p−1, bp−1). As in Section 3.1 the domain Ω is assumed to be bounded. This condition can be removed if Ω has finite measure and the boundary condition for |x| → ∞ is taken in the form (3.2.12).