Unique continuation for \(|\Delta u| \leq V|\nabla u|\) and related problems
type: Item
instance of: scholarly article
title: Unique continuation for \(|\Delta u| \leq V|\nabla u|\) and related problems
zbMATH Open document ID: 0735.35024
DOI: 10.4171/RMI/101
author: Thomas H. Wolff
published in: Revista Matemática Iberoamericana
publication date: 1992-06-25T00:00:00Z
full work available at URL: https://eudml.org/doc/39399
review text: The following theorem is proved: Theorem. Assume \(V\in L_{loc}^ r(\mathbb{R}^ d)\), \(u\in W_{loc}^{2,p}\), \(|\Delta u|\leq V|\nabla u|\) and \(\lim_{R\to 0}R^{-N}\int_{| x|\leq R}|\nabla u|^{p'}=0\) for all \(N\). Then \(u\) is a constant. As usual, \(1/p+1/p'=1\), \(1/p-1/p'=1/r\). This interesting result is a slight improvement of a theorem of Y. M. Kim. On the other hand, this paper is of interest from technical point of view as a variant of the Carleman method is proposed here and a partly new approach to proving the Carleman (i.e. weighted Sobolev) inequalities is developed.
Mathematics Subject Classification ID: 35B60
Mathematics Subject Classification ID: 35J15
Mathematics Subject Classification ID: 35R45
zbMATH DE Number: 13089
zbMATH Keywords: Carleman inequality
zbMATH Keywords: unique continuation
reviewed by: Peter R. Popivanov
MaRDI profile type: Publication
OpenAlex ID: W75324948
description: scientific article; zbMATH DE number 13089