Maximum Principles for Divergence Structure Elliptic Differential Inequalities

Abstract

For a large number of divergence structure equations, including equations which involve the important p-Laplacian operator Δ_p, there is a further series of maximum principles. In particular, in this chapter we study the differential inequality

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

(3.1.1)

where Ω is a bounded domain in ℝⁿ (unless otherwise stated explicitly), and

$$ A\left( {x,z,\xi } \right):\Omega \times \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n ,B\left( {x,z,\xi } \right):\Omega \times \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}. $$

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