Global logarithmic stability results on the Cauchy problem for anisotropic wave equations

Abstract

We discuss the Cauchy problem for anisotropic wave equations. Precisely, we address the question to know which kind of Cauchy data on the lateral boundary are necessary to guarantee the uniqueness of continuation of solutions of an anisotropic wave equation. In the case where the uniqueness holds, the natural problem that arise naturally in this context is to estimate the solutions, in some appropriate space, in terms of norms of the Cauchy data. We aim in this paper to convert, via a reduced Fourrier-Bros-Iagolnitzer transform, the known stability results on the Cauchy problem for elliptic equations to stability results on the Cauchy problem for waves equations. By proceeding in that way the main difficulty is to control the residual terms, induced by the reduced Fourrier-Bros-Iagolnitzer transform, by the Cauchy data. Also, the uniqueness of continuation from Cauchy data is obtained as byproduct of stability results.