# Numerical studies of the optimization of the first eigenvalue of the heat diffusion in inhomogeneous media

## Abstract

In this paper, we study optimization of the first eigenvalue of \(-\nabla \cdot (\rho (x) \nabla u) = \lambda u\) in a bounded domain \(\Omega \subset {\mathbb {R}}^n\) under several constraints for the function \(\rho \). We consider this problem in various boundary conditions and various topologies of domains. As a result, we numerically observe several common criteria for \(\rho \) for optimizing eigenvalues in terms of corresponding eigenfunctions, which are independent of topology of domains and boundary conditions. Geometric characterizations of optimizers are also numerically observed.

## Keywords

Eigenvalue problem Topology optimization Dependence of optimizers on topology## Mathematics Subject Classification

35Q93 49J20 65N25 74G15## Notes

### Acknowledgments

This research was partially supported by JST, CREST: A Mathematical Challenge to a New Phase of Material Science, Based on Discrete Geometric Analysis. KM was partially supported by Coop with Math Program, a commissioned project by MEXT. HN was partially supported by Grants-in-Aid for Scientific Research (C) (No. 26400067). We thank Prof. Hideyuki Azegami for providing us with very meaningful advice for our computational study. We also thank Prof. Motoko Kotani for introducing us to this study and giving us a lot of advice for writing this paper.

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