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

Abstract

In this paper, we study the scale-invariant quantity \[\mathcal{G}(Ω)=\frac{\|\partial_n u_1\|_{L^\infty(\partialΩ)}}{λ_1},\]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.

Figures (3)

• Figure 1: (a) The $L^2(\Omega)$-normalized first Dirichlet eigenfunction $u_1$ of the unit semidisk, shown using the color scale to the right. (b) Its boundary derivative plotted vs $s$, the counterclockwise arc-length from the bottom-left corner $(-1,0)$ of $\Omega$ in panel (a). The boundary function is divided by $\lambda_1$ to make it scale-invariant. Its maximum over $s$, denoted by ${\mathcal{G}}(\Omega)$, is shown as a dot in (b), and occurs at $s=1$ (corresponding to the origin in panel (a)). Conjecture \ref{['conj: semicircle']} is that there is no other convex domain $\Omega$ that exceeds this maximum of $C^*$.
• Figure 2: Optimization results with $N=16$: Each row represents a different initialization. Going from top to bottom, we have circle, square and triangle initializations. Each column is a different time step during optimization.
• Figure 3: A large scale experiment that starts with $N=8$ vertices, and gradually increases the number of vertices exceeds $N_{\rm{target}}=100$ with gradient threshold set to $\eta=5\times 10^{-6}$. Under vertex addition strategy in VertexRefinement, the resulting polygon has $N=113$ vertices.

Theorems & Definitions (25)

• Remark 1: A physical interpretation
• Theorem 2
• Conjecture 3
• Theorem 4
• Remark 5: Convexity
• proof : Proof of Theorem \ref{['thm: loose upper bound']}
• Lemma 6
• proof
• Lemma 7
• proof