Approximation of the conductivity coefficient in the heat equation

Abstract

In this paper, we study the conductivity coefficient determination in the heat equation from observation of the lateral Dirichlet-to-Neumann map. We define a bilinear form function Q_γ associated to the boundary condition and the Dirichlet-to-Neumann map, and prove that the linearized problem d Q_γ is injective. Based on the idea of complex geometrical optics solutions, we give two approximations to the conductivity coefficient by using the Fourier truncation method and the mollification method. Under the a priori assumption of the conductivity, we estimate the errors between the conductivity coefficient and its approximations by setting a suitable bound of the frequency.

Keywords:

• heat equation
• Dirichlet-to-Neumann map
• coefficient identification
• Fourier truncation method
• mollification method

AMS Subject Classifications:

• 34K29
• 34M50
• 35K65

Acknowledgements

This work has been done during the first author’s visit to the Department of Mathematics in University of Washington. It’s a pleasure to thank Prof Gunther Uhlmann and Dr Yang Yang for many value discussion and comments. The authors are also very grateful to the anonymous referees for their valuable comments, most of which are reflected in the final version. This work was supported by the National Natural Science Foundation of China [grant number 11171054].