Unique continuation for parabolic operators
type: Item
instance of: scholarly article
title: Unique continuation for parabolic operators
zbMATH Open document ID: 1028.35052
DOI: 10.1007/BF02384566
author: L. Escauriaza
author: F. J. Fernández
published in: Arkiv för Matematik
publication date: 2003-12-15T00:00:00Z
review text: Unique continuation results for elliptic and parabolic operators have been proved by many authors under various assumptions and using a variety of techniques. Here the authors consider this problem for parabolic operators of the form \[ Pu = \sum_{i,j=1}^n \partial_i \left( a^{ij}(x,t)\partial_ju \right) + \partial_tu \tag{1} \] where \(a(x,t)=[a^{ij}(x,t)]\) is symmetric with eigenvalues bounded between \(\lambda\) and \(1/\lambda\) for some \(\lambda\in(0,1]\). The main result is that if there are constants \(M>0\) and \(\beta\in(0,1)\) such that either \[ |a(x,t)-a(y,s)|\leq M(|x-y|^2+|t-s|)^{1/2} \] or \[ \begin{aligned}|a(x,t)-a(y,s)|&\leq M(|x-y|^2+|t-s|)^{\beta/2},\\ |\nabla a(x,t)|&\leq M|x|^{\beta-1}, \\ |\partial_t a(x,t)|&\leq Mt^{\beta/2-1}, \end{aligned} \] for all \((x,t),(y,s)\in{\mathbb{R}}^{n+1}\), and \(u\) satisfies \[ |Pu|\leq M(|\nabla u|+|u|) \quad\text{in}\quad B_2\times[0,2] \] and \[ |u(x,t)|\leq C_k(|x|+\sqrt{t})^k \] for all nonnegative integers \(k\), then \(u(x,0)=0\) for all \(x\in B_2\). Furthermore, \(u\) satisfies the estimate \[ |u(x,t)|\leq Ne^{-1/Nt} \|u\|_{L^\infty}(B_2\times(0,2)) \quad\text{for}\quad (x,t)\in B_1\times(0,1) \] where \(N\) depends only on \(\lambda,\beta,n\) and \(M\). The results are also proved up to the boundary under suitable Dirichlet or Neumann boundary conditions, extending earlier work of \textit{L.~Escauriaza} and \textit{V.~Adolfsson } in [Commun. Pure Appl. Math. 50, 935-969 (1997; Zbl 0899.31004)] on parabolic operators with time independent coefficients. Examples of \textit{K.~Miller} in [Arch. Ration. Mech. Anal. 54, 105-117 (1963; Zbl 0289.35046)] and \textit{A.~Plis} in [Bull. Acad. Polon. Sci., Math. Astron. Phys. 11, 95-100 (1963; Zbl 0107.07901)] show that the Lipschitz regularity of the coefficients in the space variables is necessary, but is is not known whether the \(1/2\)-Hölder regularity with respect to time is optimal. The result is derived from a Carleman type inequality. A similar app…
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35R45
zbMATH DE Number: 2017825
reviewed by: John Urbas
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2017825