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

TL;DR

This paper numerically investigates how to optimize the first eigenvalue of heat diffusion in inhomogeneous media across different domain topologies and boundary conditions, revealing common criteria and geometric features of optimal conductivities.

Contribution

It provides a comprehensive numerical analysis of eigenvalue optimization in heat diffusion, identifying universal criteria and geometric properties of optimal conductivities across various settings.

Findings

Common criteria for conductivity optimization identified

Geometric characteristics of optimal conductors observed

Results are consistent across different domain topologies and boundary conditions

Abstract

In this paper, we study optimization of the first eigenvalue of the heat equation with spatially nonuniform conductivity on a bounded domain under several constraints for the conductivity. We consider this problem in various boundary conditions and various type of topology of domains. As a result, we numerically observe several common criteria of the conductivity for optimizing eigenvalues in terms of corresponding eigenfunctions, which are independent of topology of domains and boundary conditions. The geometric characterization of optimizers are also numerically observed.