Hot Proofs: Vector-Field Methods in Eigenfunction Analysis

• Hot proofs are a rigorous framework applying vector-field variational principles to confirm that Neumann eigenfunctions attain their extrema exclusively on domain boundaries.
• This approach replaces classical scalar Rayleigh quotient techniques with streamlined proofs that integrate boundary curvature effects to unify Neumann and Dirichlet cases.
• The method employs innovative comparison techniques in vector-valued function spaces, yielding robust insights into the geometric influence on eigenvalue distributions.

Hot proofs concern rigorous approaches to the classical "hot spots" conjecture, which studies the location of extrema of Laplacian eigenfunctions under Neumann boundary conditions on bounded, simply connected planar domains. The central assertion: nonconstant eigenfunctions associated to the first nontrivial Neumann eigenvalue achieve their maximum and minimum values exclusively on the boundary of the domain. Recent developments leverage non-standard variational principles involving vector-fields, contrasting traditional scalar Rayleigh quotient frameworks and yielding streamlined proofs for families of domains. These methodologies unify Neumann and Dirichlet cases, elucidate boundary effects via curvature, and replace probabilistic and perturbative techniques with comparison in vector-valued function spaces (Rohleder, 2 Apr 2024).

1. The Classical Hot Spots Conjecture

Given a bounded, simply connected domain $\Omega \subset \mathbb{R}^2$, the classical "hot spots" conjecture posits that every eigenfunction $u \in H^1(\Omega)$ of the Laplacian $-\Delta u = \lambda_1^N u$ with Neumann boundary conditions $\partial u / \partial n = 0$ on $\partial \Omega$, and $\lambda_1^N$ denoting the first nontrivial (positive) Neumann eigenvalue, attains its maximum and minimum on the boundary: $$\max_{x \in \Omega} u(x) = \max_{x \in \partial \Omega} u(x), \qquad \min_{x \in \Omega} u(x) = \min_{x \in \partial \Omega} u(x).$$ This property encapsulates the localization of extremal "hot spots" on the domain's perimeter, excluding interior critical points except for constant solutions.

2. Standard and Non-Standard Variational Principles

2.1. Scalar Rayleigh Quotients

Traditional approaches employ scalar Rayleigh quotients. For $\Omega$ as above,

• The first nontrivial Neumann eigenvalue is given by: $$\lambda_1^N = \min_{v \in H^1(\Omega),\, \int_\Omega v = 0} \frac{\int_\Omega |\nabla v|^2 dx}{\int_\Omega |v|^2 dx}$$
• The first Dirichlet eigenvalue is: $$\lambda_1^D = \min_{v \in H_0^1(\Omega) \setminus \{0\}} \frac{\int_\Omega |\nabla v|^2 dx}{\int_\Omega |v|^2 dx}$$

2.2. Rohleder's Vector-Field Principle

Recent proofs implement a non-standard variational principle. One defines the quadratic form: $$a[u] := \int_\Omega \left( |\nabla u_1|^2 + |\nabla u_2|^2 \right) dx - \int_{\partial \Omega} \kappa(s)\,(u_1^2 + u_2^2)\,ds$$ where $u = (u_1, u_2) \in H_N$, the domain of vector-fields tangent to the boundary ($u \cdot n = 0$ almost everywhere on $\partial \Omega$), and $\kappa(s)$ is the signed boundary curvature.

The associated self-adjoint operator $A$ in $L^2(\Omega)^2$ yields the union of all positive Neumann and Dirichlet eigenvalues. The crucial variational characterization is: $$(★) \qquad \lambda_1^N = \min_{u \in H_N \setminus \{0\}} \frac{ a[u] }{ \|u\|^2_{L^2(\Omega)^2} }$$ with optimizer $u = \nabla \varphi_1^N$, directly linking the Neumann eigenfunction and its gradient structure.

The Dirichlet case appears as the next eigenvalue: $$(†) \qquad \lambda_1^D = \min_{u \in H_N,\,\, u \perp \nabla \varphi_1^N} \frac{ a[u] }{ \|u\|^2_{L^2(\Omega)^2} }$$ where orthogonality is enforced with respect to the spanning gradient field.

3. Core Variational Steps and Maximum Principle

The vector-field method synthesizes several critical inequalities and identities:

• Replacing $u$ with its modulus $v(x) := (|u_1(x)|, |u_2(x)|)$ preserves the domain and yields $\|v\|_2 = \|u\|_2$, with the Dirichlet integral and boundary term obeying:

$$\int_\Omega (|\nabla v_1|^2 + |\nabla v_2|^2) \leq \int_\Omega (|\nabla u_1|^2 + |\nabla u_2|^2)$$

for suitably oriented domains (e.g., "lip-domains" after $\pi/4$ rotation, ensuring $n_1 n_2 \leq 0$ on $\partial \Omega$).

• Minimality in $(★)$ implies equality of the Rayleigh quotients, ensuring each component attains the same variational minimum and obeys the Euler–Lagrange equation:

$$-\Delta v_j = \lambda_1^N v_j \quad \text{in } \Omega,\qquad n \cdot \nabla v_j = 0 \quad \text{on } \partial \Omega$$

for each $j = 1,2$.

• Applying the strong maximum principle to $v_j \geq 0$ (superharmonic for $\lambda_1^N > 0$), one deduces each $v_j$ is either strictly positive or vanishes identically. If $v_j \equiv 0$, geometric constraints enforce degeneracy (e.g., reducing to one-dimensional dependence only possible for rectangles).

4. Streamlined Proofs in Canonical Domains

4.1. Lip-Domain Results

For lip-domains $$\Omega = \{ (x, y) : f_1(x) < y < f_2(x),\ x \in (a, b)\}$$ with Lipschitz graphs of slope $\leq 1$, rotating by $+\pi/4$ ensures the normal satisfies $n_1 n_2 \leq 0$ almost everywhere. Application of the above variational principle demonstrates strict positivity of both partial derivatives $\partial \varphi / \partial x$ and $\partial \varphi / \partial y$ in the interior, excluding interior hot spots and verifying the conjecture. Uniqueness and simplicity of $\lambda_1^N$ follow by the impossibility of linearly independent eigenfunctions with vanishing derivatives.

4.2. Domains with Axes of Symmetry

Domains symmetric with respect to both coordinate axes, and all vertical/horizontal cross-sections as segments, allow comparison fields constructed from signed combinations of $|\nabla \varphi|$. The vector-field comparison enforces sign structure for mixed second derivatives (such as $\partial^2 \varphi / \partial x \partial y$ or $\partial^2 \varphi / \partial x^2$), vanishing on symmetry axes. This recapitulates conclusions of Jerison–Nadirashvili without appealing to Brownian motion coupling or nodal line perturbation.

5. Geometric and Analytic Features of the Vector-Field Approach

Lifting from scalar eigenfunctions $\varphi$ to their gradients $u = \nabla \varphi$ introduces curvature from the boundary through a unified quadratic form. The curvature term $\int_{\partial \Omega} \kappa |u|^2$ is non-positive for convex arcs, thus analytic manipulations such as the "take-absolute-value" trick are admissible and directly lower or preserve the quotient in step 3.1.

This vector-field formalism allows:

• Unified handling of Neumann and Dirichlet eigenvalues in one operator on $H^1(\Omega)^2$
• Replacement of probabilistic approaches (Brownian motion reflection) and perturbative nodal line tracking with a single Rayleigh–Ritz comparison in the vectorfield domain $H_N$
• Robustness in domains where scalar comparison fails due to non-monotonic boundary geometry, provided the vector field satisfies boundary tangency ($u \cdot n = 0$) and modulus transformation is admissible

A plausible implication is that such streamlined approaches may extend proof techniques to broader classes of planar domains and potentially inform extensions in higher dimensions, contingent on the existence of suitable vector-field characterizations and boundary curvature controls.

6. Summary and Perspectives

Rohleder's vector-field variational principle crystallizes a direct minimization strategy for Laplacian eigenvalues on planar domains. By focusing on the gradient structure and exploiting boundary curvature, this method achieves precise control over the location and nature of eigenfunction extrema, effectively confirming the "hot spots" conjecture in lip-domains and symmetric domains with minimal auxiliary machinery. This analytic methodology significantly simplifies prior approaches and unifies the treatment of Neumann and Dirichlet spectra under a single quadratic form (Rohleder, 2 Apr 2024).