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

1. Problem Statement

The paper investigates the behavior of the first Dirichlet eigenfunction ($u_1$) of the Laplace operator on a bounded, convex planar domain $\Omega$. Specifically, the authors study the peak boundary heat flux in the long-time limit of the heat equation.

The central quantity of interest is the scale-invariant functional $G(\Omega)$, defined as:
$$G(\Omega) = \frac{\|\partial_n u_1\|_{L^\infty(\partial\Omega)}}{\lambda_1}$$
where:

• $\lambda_1$ is the first Dirichlet eigenvalue.
• $u_1$ is the corresponding $L^2$-normalized eigenfunction.
• $\partial_n u_1$ is the inward normal derivative on the boundary $\partial\Omega$.
• $\|\cdot\|_{L^\infty}$ denotes the maximum absolute value over the boundary.

The Core Question: Among all bounded convex planar domains, which shape maximizes $G(\Omega)$?

2. Methodology

The authors employ a hybrid approach combining analytical proofs, shape calculus, and high-order numerical optimization.

A. Analytical Framework

• Green's Function Estimates: To prove upper bounds, the authors utilize the representation of $u_1$ via the Dirichlet Green's function and relate the gradient of $u_1$ to the torsion function (solution to $\Delta w = -1$).
• Shape Derivatives: To analyze critical points, the authors derive shape-derivative formulae for the eigenvalue $\lambda_1$ and the boundary normal derivative $\partial_n u_1$. They utilize layer potential theory (single and double layer potentials) to handle the boundary integral equations.
• Hadamard Formula: They apply the classical Hadamard formula for eigenvalue variation and extend it to compute the variation of the normal derivative data under boundary deformations.

B. Numerical Optimization Strategy

• Boundary Integral Equations (BIE): The eigenvalue problem is reformulated as a nonlinear eigenvalue problem for the wavenumber $k = \sqrt{\lambda}$ using the operator $\frac{1}{2}I - S'_k$ (where $S'_k$ is the normal derivative of the single-layer potential).
• Discretization: The boundary is discretized using a high-order Nyström method with 16th-order Gauss–Legendre nodes. The domain is approximated by "rounded polygons" to ensure $C^2$ smoothness required for the integral operators.
• Gradient Computation: Instead of finite differences, the authors compute exact gradients of the objective function with respect to the domain shape using the derived shape derivatives. This involves:
  • Solving a linear system for the variation of the density $\delta\sigma$.
  • Computing the variation of the eigenvalue $\delta k$ via a solvability condition involving the left null-vector.
  • Using the Fast Multipole Method (FMM) and recursive skeletonization (via the chunkIE and FLAM libraries) to accelerate matrix-vector products.
• Optimization Algorithm: A gradient ascent algorithm is employed with a vertex refinement strategy. The optimization starts with a coarse polygon (e.g., $N=8$ vertices) and iteratively refines the mesh by inserting vertices between existing ones as the gradient tolerance is met, allowing the shape to evolve from simple polygons to complex curves.

3. Key Contributions

A. Theoretical Results

1. Universal Upper Bound (Theorem 2): The authors prove that for any bounded convex domain $\Omega$, there exists a domain-independent constant $C$ such that:
   $$\|\partial_n u_1\|_{L^\infty(\partial\Omega)} \le C \lambda_1$$
   This establishes that the peak flux is bounded relative to the eigenvalue, preventing blow-up in convex geometries.
2. Critical Point Analysis (Theorem 4): They prove that the semidisk is a critical point of the functional $F(\Omega)$ (a localized version of $G$) under infinitesimal perturbations of its circular arc, provided the straight diameter remains fixed. This provides analytical support for the semidisk being an extremizer.

B. The Conjecture

Based on extensive numerical evidence, the authors propose Conjecture 3:

The semidisk maximizes $G(\Omega)$ over the set of all bounded convex planar domains.

The conjectured optimal constant is:
$$C^* = \frac{1}{\sqrt{\pi} j_{1,1} |J_0(j_{1,1})|} \approx 0.365584$$
where $j_{1,1}$ is the first positive root of the Bessel function $J_1$.

C. Numerical Evidence

• The optimization algorithm consistently converges to a semidisk shape regardless of the initial geometry (tested with circles, squares, and triangles).
• The peak flux is observed to occur at the center of the straight diameter (the origin in their coordinate system), which is a self-focal point for billiard rays in the high-frequency limit, though the authors note this is a coincidence for the fundamental mode ($k=1$).

4. Results

• Optimal Shape: The numerical experiments strongly suggest the semidisk is the global maximizer.
• Comparison with Rectangles: The authors analyzed rectangular domains and found the optimal aspect ratio to be $\sqrt{3}$, yielding a value of $F \approx 0.3628$, which is slightly lower than the semidisk's $0.3656$.
• Non-Convexity: The paper notes that the functional is unbounded for non-convex domains (e.g., sectors with reentrant corners), justifying the restriction to convex domains.

5. Significance

• Physical Interpretation: The quantity $G(\Omega)$ relates to the maximum heat flux through a boundary point in the long-time limit of heat diffusion. Identifying the shape that maximizes this flux has implications for thermal engineering and material design.
• Spectral Geometry: This work addresses a fundamental problem in spectral geometry: how geometric constraints (convexity) influence the extrema of eigenfunction derivatives. It contrasts with the "Hot Spots" conjecture (Neumann boundary conditions) by focusing on Dirichlet conditions and boundary flux.
• Methodological Advancement: The paper demonstrates a robust framework for shape optimization of spectral quantities using high-order boundary integral methods and exact shape derivatives, avoiding the inaccuracies of finite-difference approximations.
• Surprising Extremizer: The result is counter-intuitive because the optimal shape (semidisk) has corners (at the junction of the arc and diameter), whereas many other spectral extremizers (like the disk for the first eigenvalue) are smooth.

6. Future Work

The authors identify several open problems:

1. Existence Proof: Proving the existence of a maximizer in a suitable topology (e.g., using the Blaschke selection theorem and continuity arguments).
2. Local Optimality: Establishing the semidisk as a strict local maximizer via second-variation analysis.
3. Generalizations: Extending the methodology to other norms of $\partial_n u_1$, higher eigenmodes, or different boundary conditions (e.g., Robin).