Kato's inequality and Kato's inequality up to the boundary
type: Item
instance of: scholarly article
title: Kato's inequality and Kato's inequality up to the boundary
zbMATH Open document ID: 1155.31004
author: Alano Ancona
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2008-10-22T00:00:00Z
review text: Let \(U\) be a domain in \(\mathbb{R}^{d}\). The classical form of \textit{T. Kato}'s inequality [Isr. J. Math. 13(1972), Proc. internat. Sympos. partial diff. Equ. Geometry normed lin. Spaces I, 135--148 (1973; Zbl 0246.35025)] says that, if \(u\) and \(\Delta u\) belong to \(L^{1}(U)\), then \(\Delta u^{+}\) is a measure and \( \Delta u^{+}\geq \mathbf{1}_{\{u\geq 0\}}\Delta u\). This has been generalized by several authors. Recently, \textit{H. Brezis} and \textit{A. C. Ponce} [Commun. Contemp. Math. 10, 1217--1241 (2008; Zbl 1162.31005)] have shown that, if \(u\in W^{1,1}(U)\), where \(U\) is bounded and smooth, and if \(\Delta u\) is a finite measure in \(U\) and the weak normal derivative \(\partial _{n}u\) is a measure on \(\partial U\), then \(\Delta u^{+}\) and \(\partial _{n}u^{+}\) are finite measures and \( \left\| \Delta u^{+}\right\| +\left\| \partial _{n}u^{+}\right\| \leq \left\| \Delta u\right\| +\left\| \partial _{n}u\right\| \). The author answers a question raised in the latter paper by showing that, under these hypotheses, \(\partial _{n}u^{+}=\mathbf{1}_{\{u>0\}}\partial _{n}u-\mathbf{1}_{\{u=0\}}(\partial _{n}u)^{-}\) when \(\{u>0\}\) and \(\{u=0\}\) are suitably defined. The proofs are potential theoretic in nature and the results are established for a class of symmetric second order elliptic operators in divergence form.
reviewed by: Stephen J. Gardiner
Mathematics Subject Classification ID: 31B25
Mathematics Subject Classification ID: 35J15
zbMATH DE Number: 5355664
zbMATH Keywords: Kato's inequality
zbMATH Keywords: Laplacian
zbMATH Keywords: boundary behaviour
zbMATH Keywords: elliptic operators
MaRDI profile type: Publication
full work available at URL: j.crma.2008.07.027
OpenAlex ID: W2046946495
cites work: Régularité d'accès des bouts et frontière de Martin d'un domaine euclidiens
cites work: Q5526430
cites work: Kato's inequality when \(\Delta u\) is a measure
cites work: KATO'S INEQUALITY UP TO THE BOUNDARY
cites work: Classical potential theory and its probabilistic counterpart.
cites work: Finely harmonic functions
cites work: Some properties of the Riesz charge associated with a \(\delta\)- subharmonic function
cites work: Q5671044
cites work: Les fonctions surharmoniques associées à un opérateur elliptique du second ordre à coefficients discontinus
cites work: Schrödinger operators with singular potentials
cites work: Sur le rôle de la frontière de R. S. Martin dans la théorie du potentiel
cites work: Q5518040
cites work: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus
DOI: 10.1016/J.CRMA.2008.07.027
Recommended article: Elliptic operators, conormal derivatives and positive parts of functions (with an appendix by Haïm Brezis)
Recommended article: Potential theory for the Schrödinger equation
Recommended article: Kato's inequality when \(\Delta u\) is a measure
Recommended article: Kato's inequality for operators of second quantization
Recommended article: Remarks on Kato's inequality when ∆pu is a measure
Recommended article: A Littlewood–Paley type inequality with applications to the elliptic Dirichlet problem
Recommended article: Positive solutions of nonlinear second-order elliptic inequalities in unbounded domains
Recommended article: Q3693962
Recommended article: Kato square root problem with unbounded leading coefficients
Recommended article: Q4005479
description: scientific article; zbMATH DE number 5355664