A unique continuation theorem for second order parabolic differential operators
type: Item
instance of: scholarly article
title: A unique continuation theorem for second order parabolic differential operators
zbMATH Open document ID: 0704.35016
DOI: 10.1007/BF02387373
published in: Arkiv för Matematik
publication date: 1990-01-01T00:00:00Z
review text: Denote by \((x,t)\) a generic point in \({\mathbb{R}}^ n\times {\mathbb{R}}\), \(n\geq 2\), and let \(X\subset {\mathbb{R}}^{n+1}\) be an open connected set. Let \(L=D_ t+P(t,x,D_ x)\) be a differential operator of parabolic type, where P is a second order elliptic operator with bounded coefficients, whose principal part is real and \(C^{\infty}.\) The author establishes the following result: Suppose that \(Lu\in L^ 2_{loc}(X)\) for some \(u\in {\mathcal D}'(X)\) and that \[ (*)\;| Lu| \leq | Vu| \text{ in } X \] for some \(V\in L_{loc}^{(n+2)/2}(X)\). Then the normal set of the support of u is non-characteristic for L. As a corollary, a unique continuation theorem is obtained: Let \(X=\Omega \times [-T,T]\), where \(\Omega \subset {\mathbb{R}}^ n\) is an open connected set and \(T>0\). Let L and V be as above and suppose that (*) holds. Assume that u vanishes in an open subset \(G\subset X\). Then u must vanish identically in the horizontal component of G, i.e., \(u(x,\tau)\in G\) for some \(\xi\). The proof of these results relies on a Carleman type inequality, the derivation of which constitutes the main part of the paper.
Mathematics Subject Classification ID: 35B60
Mathematics Subject Classification ID: 35K10
zbMATH DE Number: 4154968
zbMATH Keywords: normal set
zbMATH Keywords: non-characteristic
zbMATH Keywords: unique continuation theorem
zbMATH Keywords: Carleman type inequality
author: Christopher D. Sogge
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02387373
OpenAlex ID: W2085921661
description: scientific article; zbMATH DE number 4154968