The Classical Maximum Principle

Abstract

The purpose of this chapter is to extend the classical maximum principles for the Laplace operator, derived in Chapter 2, to linear elliptic differential operators of the form

$$ Lu = {a^{{ij}}}(x){D_{{ij}}}u + {b^i}(x){D_i}u + c(x)u, {a^{{ij = }}}{a^{{ji}}} $$

(3.1)

, where x = (x ₁,..., x _n) lies in a domain Ω of ℝⁿ, n≥2. It will be assumed, unless otherwise stated, that u belongs to C ²(Ω). The summation convention that repeated indices indicate summation from 1 to n is followed here as it will be throughout. L will always denote the operator (3.1).