Carleman estimate with second large parameter for second order hyperbolic operators in a Riemannian manifold and applications in thermoelasticity cases

Abstract

In this article we prove a Carleman estimate with second large parameter for a second order hyperbolic operator in a Riemannian manifold ℳ. Our Carleman estimate holds in the whole cylindrical domain ℳ × (0, T) independent of the level set generated by a weight function if functions under consideration vanish on boundary ∂(ℳ × (0, T)). The proof is direct by using calculus of tensor fields in a Riemannian manifold. Then, thanks to the dependency of the second larger parameter, we prove Carleman estimates also for (i) a coupled parabolic-hyperbolic system (ii) a thermoelastic plate system (iii) a thermoelasticity system with residual stress.

Keywords:

• Carleman estimates
• second large parameter
• hyperbolic equation
• thermoelasticity system

AMS Subject Classifications::

• 35B60
• 35N99
• 74F05

Acknowledgements

The authors thank the anonymous referee for very useful comments and insightful suggestions. M. Bellassoued was partly supported by UR Mathematics and Applications FSB. This work was supported partly by Global COE Program ‘The Research and Training Center for New Development in Mathematics’ of The University of Tokyo.