The peak heat flux conjecture for the first Dirichlet eigenmode of convex planar domains

TL;DR

This paper investigates the maximum boundary heat flux for the first Dirichlet eigenmode in convex planar domains, conjecturing the semidisk maximizes this quantity, supported by numerical and analytical evidence.

Contribution

It introduces a scale-invariant boundary flux measure, develops shape optimization techniques, and conjectures the semidisk maximizes this measure among convex domains.

Findings

Numerical optimization suggests the semidisk maximizes the flux.

Analytical proof shows the semidisk is a critical point of the flux.

The study provides a new conjecture linking domain shape to heat flux maxima.

Abstract

In this paper, we study the scale-invariant quantity \[\mathcal{G}(\Omega)=\frac{\|\partial_n u_1\|_{L^\infty(\partial\Omega)}}{\lambda_1},\]where $u_1$ is the first $L^2$-normalized Dirichlet Laplace eigenfunction of a Euclidean domain $\Omega$ and $\lambda_1$ is its eigenvalue. This is related to the peak boundary heat flux in the long time limit. For convex domains we prove that $\|\partial_n u_1\|_{L^\infty(\partial\Omega)}$ is upper-bounded by a (domain-independent) constant multiple of $\lambda_1$. Using layer potentials, we derive shape-derivative formulae for efficient gradient computations. When combined with high-order Nystr\"om discretization, a fast boundary integral equation solver, and eigenvalue rootfinding, this allows us to numerically optimize $\mathcal{G}$ over a class of rounded polygonal discretized domains. Based on extensive numerical experiments, we then conjecture that, over the set of convex domains, $\mathcal{G}$ is maximized by the semidisk, with the peak flux at the center of the diameter. To lend analytical support to this conjecture, we prove that the semidisk is a critical point of $\mathcal{G}$ under infinitesimal perturbations of its circular arc.