Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients
type: Item
instance of: scholarly article
title: Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients
zbMATH Open document ID: 0909.35004
author: L. Robbiano
author: Claude Zuily
published in: Inventiones Mathematicae
publication date: 1999-03-30T00:00:00Z
review text: The goal of the paper is to fill the gap between Holmgren's theorem (for linear differential operators with analytic coefficients and non-characteristic initial surface) and Hörmander's theorem (for linear differential operators with \(C^\infty\)-coefficients, where the operator is principally normal and the initial surface is pseudo-convex). A natural question is that for uniqueness results for linear differential operators \(P\) of arbitrary order with \(C^\infty\)-coefficients, where the coefficients depend analytically on \(x_a\), \(x= (x_a,x_b)\). The authors prove uniqueness results for the Cauchy problem for such operators \(P\), in this sense they fill the gap mentioned at the beginning. As usual one has to prove Carleman estimates, these are \(L^2\)-estimates with an exponential weight \(e^{-\lambda\psi}\). The assumptions of principal normality and pseudo-convexity guarantee that the operator \(P_\lambda= e^{\lambda\psi} Pe^{-\lambda\psi}\) is subelliptic. Hence Garding-type estimates can be derived. As an other important tool, the authors use the partial FBI transformation. The authors give an interesting motivation for their results from control theory. As an application, they obtain a uniqueness result for second-order strictly hyperbolic operators for any non-characteristic initial hypersurface (even time-like ones are allowed).
Mathematics Subject Classification ID: 35A07
Mathematics Subject Classification ID: 35A22
Mathematics Subject Classification ID: 35B45
zbMATH DE Number: 1143543
zbMATH Keywords: subellipticity
zbMATH Keywords: Holmgren's theorem
zbMATH Keywords: Hörmander's theorem
zbMATH Keywords: Carleman estimates
zbMATH Keywords: Garding-type estimates
zbMATH Keywords: non-characteristic initial hypersurface
reviewed by: Michael Reissig
MaRDI profile type: Publication
full work available at URL: s002220050212
OpenAlex ID: W2036669430
DOI: 10.1007/S002220050212
Recommended article: Uniqueness in the Cauchy problem for quasi-homogeneous operators with partially holomorphic coefficients
Recommended article: Uniqueness in the Cauchy problem for first order operator with C k coefficients
Recommended article: A uniqueness theorem for holomorphic families of operators
Recommended article: Q4369095
Recommended article: Q4735475
Recommended article: Uniqueness in Cauchy Problems for Hyperbolic Differential Operators
Recommended article: Q4722703
Recommended article: Q4710859
Recommended article: Uniqueness of the Cauchy problem for a class of quasihomogeneous operators.
Recommended article: Q3781448
description: scientific article; zbMATH DE number 1143543