The peak heat flux conjecture for the first Dirichlet eigenmode of convex planar domains
This addresses a theoretical problem in mathematical physics for researchers studying heat diffusion and spectral geometry, with incremental analytical support for the conjecture.
The paper tackles the problem of bounding the peak boundary heat flux for convex planar domains by studying a scale-invariant quantity related to the first Dirichlet Laplace eigenfunction, proving an upper bound and numerically conjecturing that the semidisk maximizes this quantity with the peak flux at the center of the diameter.
In this paper, we study the scale-invariant quantity \[\mathcal{G}(Ω)=\frac{\|\partial_n u_1\|_{L^\infty(\partialΩ)}}{λ_1},\]where $u_1$ is the first $L^2$-normalized Dirichlet Laplace eigenfunction of a Euclidean domain $Ω$ and $λ_1$ is its eigenvalue. This is related to the peak boundary heat flux in the long time limit. For convex domains we prove that $\|\partial_n u_1\|_{L^\infty(\partialΩ)}$ is upper-bounded by a (domain-independent) constant multiple of $λ_1$. 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.