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

In this paper, we study the scale-invariant quantity $$\mathcal{G}(\mathrm{\Omega })=\frac{\mathrm{\parallel }{\mathrm{\partial }}_n{u}_1{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathrm{\partial }\mathrm{\Omega })}}{{\lambda }_1},$$where $u_1$ is the first $L^2$-normalized Dirichlet Laplace eigenfunction of a Euclidean domain $\mathrm{\Omega }$ 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 $\mathrm{\parallel }{\mathrm{\partial }}_n{u}_1{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathrm{\partial }\mathrm{\Omega })}$ 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 Nyström 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.