Kato's Inequalities for Degenerate Quasilinear Elliptic Operators

• Journal title : Kyungpook mathematical journal
• Volume 48, Issue 1,  2008, pp.15-24
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2008.48.1.015
Title & Authors
Kato's Inequalities for Degenerate Quasilinear Elliptic Operators
Horiuchi, Toshio;

Abstract
Let $\small{N{\geq}1}$ and p > 1. Let $\small{{\Omega}}$ be a domain of $\small{\mathbb{R}^N}$. In this article we shall establish Kato's inequalities for quasilinear degenerate elliptic operators of the form $\small{A_pu}$ = divA(x,$\small{\nabla}$u) for $\small{u{\in}K_p({\Omega})}$, ), where $\small{K_p({\Omega})}$ is an admissible class and $\small{A(x,\xi)\;:\;{\Omega}{\times}\mathbb{R}^N{\rightarrow}\mathbb{R}^N}$ is a mapping satisfying some structural conditions. If p = 2 for example, then we have $\small{K_2({\Omega})\;= \;\{u\;{\in}\;L_{loc}^1({\Omega})\;:\;\partial_ju,\;\partial_{j,k}^2u\;{\in}\;L_{loc}^1({\Omega})\;for\;j,k\;=\;1,2,{\cdots},N\}}$. Then we shall prove that $\small{A_p{\mid}u{\mid}\;\geq}$ (sgn u) $\small{A_pu}$ and $\small{A_pu^+\;\geq\;(sgn^+u)^{p-1}\;A_pu}$ in D'($\small{{\Omega}}$) with $\small{u\;\in\;K_p({\Omega})}$. These inequalities are called Kato's inequalities provided that p = 2. The class of operators $\small{A_p}$ contains the so-called p-harmonic operators $\small{L_p\;=\;div(\mid{{\nabla}u{\mid}^{p-2}{\nabla}u)}$ for $\small{A(x,\xi)={\mid}\xi{\mid}^{p-2}\xi}$.
Keywords
Kato's inequality;p-harmonic operators;
Language
English
Cited by
References
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