Non-homogeneous Divergence Structure Inequalities

Abstract

We consider the quasilinear differential inequality

$$ divA\left( {x,u,Du} \right) + B\left( {x,u,Du} \right) \geqslant 0in\Omega , $$

(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,

$$ \begin{gathered} \left\langle {A\left( {x,z,\xi } \right),\xi } \right\rangle \geqslant \left| \xi \right|^p - a_2 z^p , \hfill \\ B\left( {x,z,\xi } \right) \leqslant b_1 \left| \xi \right|^{p - 1} + b_2 z^{p - 1} + b^{p - 1} , \hfill \\ \end{gathered} $$

(6.1.2)

while for p = 1,

$$ \left\langle {A\left( {x,z,\xi } \right),\xi } \right\rangle \geqslant \left| \xi \right| - a_2 z - a,B\left( {x,z,\xi } \right) \leqslant b $$

(6.1.3)

(in (6.1.3) we write b for b₂ and discard the terms b₁|ξ|^p−1, b^p−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).