Heat Transfer Shape Optimization: Stability and Non-Optimality of the Ball
Abstract
This paper investigates shape optimization problems in the context of heat transfer, with a focus on the stability and non-optimality of round domains under Robin boundary conditions. Using the flow approach and Steklov eigenvalue estimates, we derive the necessary and sufficient stability conditions for a ball to maximize the averaged heat when the heat source is radially decreasing. Our results show that, counterintuitively, a ball may not be optimal for maximizing the averaged heat under heat convection, even with radially decreasing heat sources located on the center of the ball. Moreover, we identify stability-breaking phenomena by giving precise values of thresholds, which depend on the Robin coefficient, dimension, and volume constraints. Additionally, we demonstrate that a ball can maximize the averaged temperature under certain conditions and we also explore optimal shapes in thin insulation problems.
Keywords
Cite
@article{arxiv.2307.09303,
title = {Heat Transfer Shape Optimization: Stability and Non-Optimality of the Ball},
author = {Qinfeng Li and Hang Yang},
journal= {arXiv preprint arXiv:2307.09303},
year = {2025}
}