New realization of the pseudoconvexity and its application to an inverse problem

Abstract

We consider a hyperbolic differential operator with variable principal term. We first give a sufficient condition for the pseudoconvexity which yields a Carleman estimate and a necessary condition. Our sufficient condition implies that level sets generated by the weight function in the Carleman estimate are convex with respect to the set of rays given by a ₀(x), and give a more general explicit condition of a ₀ for the pseudoconvexity. Second we apply the Carleman estimate to an inverse problem of determining a ₀ by Cauchy data on a lateral boundary with relaxed constraints on a ₀.

Keywords:

• Carleman estimate
• pseudoconvexity
• convexity
• ray
• inverse problem

AMS Subject Classifications:

• 35B60
• 35L15
• 35R30

Acknowledgements

The authors thank Professor Mourad Bellassoued for valuable comments. Oleg Imanuvilov was supported in part by NSF Grant DMS 0808130 and Victor Isakov was supported in part by NSF Grant DMS 04-05976, 07-07734.