Kato's Inequalities for Degenerate Quasilinear Elliptic Operators

Abstract

Abstract. Let N ≥ 1 and p > 1. Let Ω be a domain of R N . In this arti-cle we shall establish Kato’s inequalities for quasilinear degenerate elliptic operators ofthe form A p u = divA(x,∇u) for u ∈ K p (Ω), where K p (Ω) is an admissible class andA(x,ξ) : Ω×R N → R N is a mapping satisfying some structural conditions. If p = 2 for ex-ample, then we have K 2 (Ω) = {u ∈ L 1loc (Ω) : ∂ j u,∂ 2j,k u ∈ L 1 (Ω) for j,k = 1,2,··· ,N}.Then we shall prove that A p |u| ≥ (sgnu)A p u and A p u + ≥ (sgn + u) p−1 A p u in D 0 (Ω) withu ∈ K p (Ω). These inequalities are called Kato’s inequalities provided that p = 2. Theclass of operators A p contains the so-called p−harmonic operators L p = div(|∇u| p−2 ∇u)for A(x,ξ) = |ξ| p−2 ξ. 1. IntroductionLet N ≥ 1and p > 1. Let Ω be a domain of R N . In this paper we shall considerthe operators of the form(1.1) A p u = divA(x,∇u),where A : Ω×R N → R N is a mapping satisfying the following conditions for somepositive constants C 1 , C 2 and C 3 :(1.2) A(x,ξ) ∈ C