Maximum and Comparison Principles
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.
KeywordsMaximum Principle Comparison Principle Poincare Inequality Minimal Surface Equation Weak Maximum Principle
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