Unique continuation for a class of degenerate elliptic operators
type: Item
instance of: scholarly article
title: Unique continuation for a class of degenerate elliptic operators
zbMATH Open document ID: 0778.35013
DOI: 10.1016/0022-247X(92)90176-E
author: Jun Ling
published in: Journal of Mathematical Analysis and Applications
publication date: 1993-04-01T00:00:00Z
review text: The paper is concerned with the strong unique continuation property for the system of degenerate elliptic equations \[ \text{div}(| u|^{p-2}\nabla u^{(k)})-f^{(k)}(x,u)=0\quad\text{for } x\in\Omega,\;k=1,\ldots,s. \tag{*} \] Here \(\Omega\subset\mathbb{R}^ n\) \((n\geq 3)\) is a domain, \(s\in\mathbb{N}\), \(p\geq 2\). On \(f:\Omega\times\mathbb{R}^ s\to\mathbb{R}^ s\) the growth condition \(| f(x,t)|\leq K| t|^{p-1}\) is imposed. The author presents the following result: Let \(u\in W^{2,2}_{\text{loc}}(\Omega,\mathbb{R}^ s)\) be a solution to (*). Then \(u\) cannot vanish of infinite order at some \(x_ 0\in\Omega\) unless \(u\) vanishes identically in \(\Omega\). Reviewer's remark: The solution class \(W^{2,2}_{\text{loc}}\) is not adequate for the degenerate system. The author should have given some information or reference concerning the natural solution class for (*). In the case of systems \((s>1)\) the proof contains some gaps. In the case of equations \((s=1)\) by means of the substitution \(w:=| u|^{p-2}u\) the problem essentially reduces to a linear one, which has been treated e.g. by \textit{N. Garofalo} and \textit{F.-H. Lin} [Indiana Univ. Math. J. 35, 245-268 (1986; Zbl 0678.35015)].
Mathematics Subject Classification ID: 35B60
Mathematics Subject Classification ID: 35J70
zbMATH DE Number: 150612
zbMATH Keywords: strong unique continuation
zbMATH Keywords: degenerate elliptic equations
reviewed by: Hans-Christoph Grunau
MaRDI profile type: Publication
cites work: Q3834387
cites work: Unique continuation for elliptic operators: A geometric-variational approach
cites work: Uniqueness theorems for second order elliptic differential equations
cites work: Unique continuation and absence of positive eigenvalues for Schrödinger operators. (With an appendix by E. M. Stein)
full work available at URL: 0022-247x(92)90176-e
OpenAlex ID: W1976499704
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description: scientific article; zbMATH DE number 150612