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

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.