The peak heat flux conjecture for convex planar domains

Let $\Omega$ be a bounded convex domain in the plane, let $\lambda_1$ be its first Dirichlet eigenvalue, and let $u_1$ be the corresponding $L^2$-normalized eigenfunction. Define

$$\mathcal{G}(\Omega):=\frac{\|\partial_n u_1\|_{L^{\infty}(\partial\Omega)}}{\lambda_1}.$$

Let $J_n$ denote the Bessel function of the first kind of order $n$, and let $j_{n,i}$ be the $i$-th positive root of $J_n$. The peak heat flux conjecture. If $\mathcal{D}$ is the space of all bounded convex domains in the plane, then

$$\sup_{\Omega\in\mathcal{D}}\mathcal{G}(\Omega)=\max_{\Omega\in\mathcal{D}}\mathcal{G}(\Omega)=C^*,$$

where

$$C^*:=\frac{1}{\sqrt{\pi}j_{1,1}|J_0(j_{1,1})|}\approx 0.3655840228073865,$$

and the maximum is attained when $\Omega$ is a semidisk. The conjecture concerns the largest scale-invariant peak boundary heat flux among convex planar domains; the paper proves a domain-independent upper bound of the form $\|\partial_n u_1\|_{L^\infty(\partial\Omega)}\leq C\lambda_1$ and supplies numerical evidence for the semidisk extremizer, while the asserted sharp maximum remains open.