Determination of a coefficient in the wave equation with a single measurement

Abstract

We consider an inverse problem of finding the coefficient of the second-order derivatives in a second-order hyperbolic equation with variable coefficients. Under a weak regularity assumption and a geometrical condition on the metric, we prove the uniqueness in a multidimensional hyperbolic inverse problem with a single measurement of Neumann data on a suitable sub-boundary. Moreover we show that our uniqueness yields the Lipschitz stability estimate in L ² space for solution to the inverse problem. The key is a Carleman estimate for a hyperbolic operator with variable coefficients.

Keywords:

• Carleman estimates
• inverse problems

Acknowledgements

The work by Masahiro Yamamoto was supported partially by Grant 15340027 from the Japan Society for the Promotion of Science and Grant 15654015 from the Ministry of Education, Cultures, Sports and Technology. The authors thank anonymous referees for invaluable comments.