# Maximum and Comparison Principles

• David Gilbarg
• Neil S. Trudinger
Part of the Classics in Mathematics book series (volume 224)

## 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)Diju + b(x, u, Du), a ij = a ji , where x = (x1..., xn) is contained in a domain Ω of ℝ n , n ≥ 2, and, unless other-wise stated, the function u belongs to C2(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
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