On the uniqueness of the Cauchy problem under partial analyticity assumptions. (English) Zbl 0907.35002

Hörmander, Lars

Colombini, Ferruccio (ed.) et al., Geometrical optics and related topics. Selected papers of the meeting, Cortona, Italy, September 1996. Boston, MA: Birkhäuser. Prog. Nonlinear Differ. Equ. Appl. 32, 179-219 (1997).

This paper deals with the uniqueness of the solutions of the Cauchy problem for partial differential operators. The author proves some uniqueness theorems for operators with partially analytic coefficients, i.e. with coefficients which are analytic with respect to only some of the variables. These results improve the previous works of R. Robbiano [Commun. Partial Differential Equations 16, 789-800 (1991; Zbl 0735.35086)], L. Hörmander [Commun. Partial Differential Equations 17, 699-714 (1992; Zbl 0815.35063)] and O. Tataru [Commun. Partial Differential Equations 20, 855-884 (1995; Zbl 0846.35021)], and make a link between Holmgren’s uniqueness theorem for operators with analytic coefficients and Hörmander’s uniqueness theorem for principally normal operators with \(C^\infty\) principal part, under the hypothesis of strong pseudoconvexity of the surface of initial data.
Denoting by \(x=(x', x'')\) the variables in \({\mathbb R}^n\) and by \(\xi=(\xi', \xi'')\) the corresponding dual variables, and supposing that all the coefficients of the operator \(P(x,D)\) are analytic with respect to \(x'\), uniqueness across the surface \(\{\varphi(x)=0\}\) is shown under the following assumptions which involve only cotangent vectors with \(\xi'=0\): (i) a transversal principal normality condition on \(P\); (ii) a pseudoconvexity condition on the surface \(\{\varphi(x)=0\}\); (iii) a condition on the principal symbol \(p\) of the operator \(P\), i.e. \(p(x', x'', 0, \xi'')\) is independent of \(x'\). In the case of transversally elliptic operators the assumption (iii) can be dropped.
The technique of the proofs is based on a modification of the Carleman estimates method, first used by Tataru.
For the entire collection see [Zbl 0878.00061].

Reviewer: Daniele Del Santo (Trieste)

MSC:

35A07	Local existence and uniqueness theorems (PDE) (MSC2000)
35L10	Second-order hyperbolic equations

Keywords:

Holmgren’s uniqueness theorem; Hörmander’s uniqueness theorem; Carleman estimate; pseudoconvexity of the surface of initial data

Citations:

Zbl 0735.35086; Zbl 0815.35063; Zbl 0846.35021