Conditional Stability for an Inverse Neumann Boundary Problem

Abstract

In this article, we consider an inverse problem of determining an unknown boundary in where the zero Neumann condition is imposed. We prove the uniqueness and a stability estimate under some a priori assumptions on unknown boundaries and the solutions of the problems. The proofs are based on the complex extension method and an estimation of harmonic measure. One of the advantages of our method is that we need not pose the boundary condition on the whole boundary, which is quite practical.

Keywords:

• Determining unknown boundary
• Neumann boundary condition
• Conditional stability estimate

Acknowledgments

The first author is partially supported by the NSF of China (No. 10271032). The third author is supported partially by Grants 15340027 and 15654015 from the Japan Society for the Promotion of Science and the Sanwa Systems Development Co. Ltd. (Tokyo, Japan). The work described in this article was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (RCG Ref: CityU 1178/02P).

Notes

^†E-mail: [email protected]

*E-mail: [email protected]

*E-mail: [email protected]

Additional information

Notes on contributors

M. Yamamoto

†E-mail: [email protected] *E-mail: [email protected]