# Maximum and Comparison Principles

## Abstract

The purpose of this chapter is to provide various maximum and comparison principles for quasilinear equations which extend corresponding results in Chapter 3. We consider second order, quasilinear operators *Q* of the form (10.1) *Qu* = *a*^{ ij }(*x*, *u*, *Du*)*D*_{ij}*u* + *b*(*x*, *u*, *Du*), *a*^{ ij } = *a*^{ ji }, where *x* = (*x*_{1}..., *x*_{n}) is contained in a domain *Ω* of ℝ^{ n }, *n* ≥ 2, and, unless other-wise stated, the function *u* belongs to *C*^{2}(*Ω*). The coefficients of *Q*, namely the functions *a*^{ ij }(*x*, *z*, *p*), *i, j*= 1,..., *n*, *b*(*x*, *z*, *p*) are assumed to be defined for all values of (*x*, *z*, *p*) in the set *Q* × ℝ × ℝ^{ n }. Two operators of the form (10.1) will be called *equivalent* if one is a multiple of the other by a fixed positive function in *Q* × ℝ × ℝ^{ n }. Equations *Qu* = 0 corresponding to equivalent operators *Q* will also be called equivalent.

## Keywords

Maximum Principle Comparison Principle Poincare Inequality Minimal Surface Equation Weak Maximum Principle## Preview

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