Non-homogeneous Divergence Structure Inequalities

Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 73)

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, 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).

Keywords

Maximum Principle Lebesgue Space Curvature Equation Structure Inequality Homogeneous Linear Equation
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