Unique continuation on a line for the Helmholtz equation

Abstract

In this article, local unique continuation on a line for solutions of the Helmholtz equation is discussed. The fundamental solution of the exterior problem for the Helmholtz equation have a logarithmic singularity which behaves similar to those of the interior problem for the Laplace equation in two dimension. A Hölder-type conditional stability estimate of the proposed exterior problem for the Helmholtz equation is obtained by adopting the complex extension method in Cheng and Yamamoto [J. Cheng and M. Yamamoto, Unique continuation on a line for harmonic functions, Inverse Probl. 14 (1998), pp. 869–882]. Finally, a regularization scheme based on the collocation method is compatible with the Hölder-type stability estimate provided that the line does not intersect the boundary of the domain for both the Laplace and the Helmholtz equations.

Keywords:

• unique continuation
• line continuation
• Helmholtz equation

AMS Subject Classifications::

• 65N21
• 35J05

Acknowledgements

S. Lu is supported by the Shanghai Science and Technology Commission Grant: 11ZR1402800. B. Xu and X. Xu are partly supported by NSFC (no 10871050) and Innovative Interdisciplinary Training Program of Fudan University (no EHH1140201).