Asymptotic behavior for the principal eigenvalue of a reinforcement problem

ABSTRACT

In this paper, we consider the asymptotic behavior for the principal eigenvalue of an elliptic operator with piecewise constant coefficients. This problem was first studied by Friedman in 1980. We show how the geometric shape of the interface affects the asymptotic behavior for the principal eigenvalue. This is a refinement of the result by Friedman.

KEYWORDS:

• Eigenvalue problem
• two phase
• transmission condition
• reinforcement problem
• domain perturbation

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35J20
• 49R05

Acknowledgements

The author would like to thank Professor Shigeru Sakaguchi (Tohoku University) for many stimulating discussions. Also the author would like to thank Lorenzo Cavallina (Tohoku University) for his warm encouragement.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This research was partially supported by the Grant-in-Aid for Scientific Research (B) [#26287020] and Challenging Exploratory Research [#16K13768] of Japan Society for the Promotion of Science.