Where the slowest heat mode reaches its hottest and coldest values

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Family 369 · Partial differential equations
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-26

1. The question

Take a thin insulated plate with an uneven initial temperature. As heat diffuses, the temperature approaches its spatial average. The slowly decaying part of the remaining temperature variation is described by the lowest nonconstant Neumann eigenfunctions.

Must the hottest and coldest points of such an eigenfunction lie on the plate’s boundary? This is the spectral hot-spots problem associated with Rauch.

Family 369 claims a positive answer for every bounded smooth simply connected planar domain, with no convexity or symmetry assumption. It claims more: the gradient of every nonzero first-positive Neumann eigenfunction is nonzero everywhere in the interior. Thus the result would exclude interior saddle points as well as interior maxima and minima.

The statement applies even when the eigenvalue is multiple. It concerns every nonzero eigenfunction in the relevant eigenspace, rather than the possibility of choosing one particularly favorable eigenfunction.

2. Background and essential definitions

Insulation leads to Neumann boundary conditions

Let $\Omega\subset\mathbb R^2$ be the plate. With thermal diffusivity normalized to one, temperature $T$ satisfies

$$\partial_tT=\Delta T, \qquad \partial_\nu T=0\quad\text{on }\partial\Omega.$$

Here $\nu$ is the outward unit normal. The zero normal derivative expresses insulation: no heat crosses the boundary.

Separation of variables leads to the eigenvalue problem

$$-\Delta u=\mu u\quad\text{in }\Omega, \qquad \partial_\nu u=0\quad\text{on }\partial\Omega.$$

The constant functions have eigenvalue zero. The first positive eigenvalue is the smallest $\mu>0$ for which a nonzero solution exists. This digest denotes it by $\mu_1$; conventions that count the constant mode first often call the same eigenvalue $\mu_2$.

Its eigenfunctions have mean zero and therefore take both signs. Variationally,

$$\mu_1=\inf_{\substack{v\in H^1(\Omega),\ v\ne0\\ \int_\Omega v=0}} \frac{\int_\Omega|\nabla v|^2\,dx} {\int_\Omega|v|^2\,dx}.$$

The space $H^1$ consists of square-integrable functions with square-integrable weak first derivatives. The quotient measures gradient energy relative to the size of the function.

Why the first mode describes late-time behavior

An eigenfunction contributes a term $e^{-\mu t}u(x)$ to heat flow. Among nonconstant modes, the first positive eigenspace decays slowest. If the initial temperature has a nonzero projection onto that space, the leading deviation from the average at large times is a fixed eigenfunction from it.

If that projection vanishes, a higher mode leads instead. Also, an asymptotic statement about the leading mode is not automatically a theorem locating the exact hottest point at every finite time. The manuscript’s theorem is stated for eigenfunctions.

Three distinctions in the conjecture

An interior point $x$ is critical if $\nabla u(x)=0$. A local extremum is critical, but a saddle point can be critical without being an extremum.

The nonstrict hot-spots statement permits an interior value tied with the global boundary maximum or minimum. The strict statement says every interior value lies strictly between the two boundary extremes. Nonvanishing gradient is stronger still because it excludes saddle points.

A domain is simply connected if every closed loop can be contracted to a point within the domain. For a smooth bounded planar region, this means there are no holes. It permits severe indentations and narrow passages; it does not imply convexity.

Multiplicity means that several linearly independent eigenfunctions share $\mu_1$. Any linear combination remains an eigenfunction. A result for one basis vector would not settle the statement for the entire eigenspace.

3. Development history and previous partial results

Rauch discussed the problem in connection with long-time heat flow in the 1970s. Bañuelos and Burdzy later distinguished strict and nonstrict formulations and the quantifier over eigenfunctions, distinctions that become essential when multiplicities occur.

Topology soon proved important. Burdzy and Werner constructed a planar counterexample with two holes and a simple first positive eigenvalue. Bass and Burdzy obtained examples with both extrema strictly inside. Burdzy’s one-hole construction showed that even one hole can allow failure. The remaining simply connected planar assertion is therefore a substantive topological restriction.

Positive results developed under geometric assumptions. Jerison and Nadirashvili obtained a positive directional derivative for every relevant eigenfunction on convex planar domains with two perpendicular reflection symmetries. Pascu treated antisymmetric eigenfunctions on suitable convex domains with one reflection symmetry. Atar and Burdzy treated lip domains, bounded between graphs with Lipschitz constants at most one. Miyamoto excluded interior critical points under a spectral-diameter condition, including nonsymmetric nearly circular convex examples.

Polygons have their own theory. Siudeja treated a class of acute triangles. Judge and Mondal, together with their subsequent erratum, excluded interior critical points for every first-positive Neumann eigenfunction on every Euclidean triangle. Chen, Gui and Yao later gave a finer classification of nonvertex critical points. These corner-domain results are distinct from the smooth-domain theorem reported here.

Recent quantitative work, including de Dios Pont, Hsu and Taylor, bounds the ratio between an eigenfunction’s supremum over a domain and its boundary maximum. Such estimates measure possible failure of hot spots but do not by themselves exclude saddle points.

The main methodological starting point is Rohleder’s variational approach. It treats eigenfunction gradients as tangent vector fields and gives an analytic route to hot spots on lip domains. The manuscript seeks a replacement for the preferred directions available in those geometries, using scalar boundary multipliers and Green kernels.

4. What the new manuscript claims

Let $\Omega\subset\mathbb R^2$ be a nonempty bounded simply connected open set with $C^\infty$ boundary. Let $V$ be the real eigenspace for its first positive Neumann eigenvalue.

The main theorem states that for every $0\ne u\in V$,

$$\nabla u(x)\ne0\qquad\text{for every }x\in\Omega.$$

Consequently,

$$\min_{y\in\partial\Omega}u(y) <u(x) <\max_{y\in\partial\Omega}u(y) \qquad(x\in\Omega).$$

All global maxima and minima over the closed domain therefore lie on its boundary. Smoothness supplies a continuous boundary extension, so these extrema are attained.

No convexity, reflection symmetry, preferred direction, eigenvalue simplicity or small deformation from a standard domain is assumed. The conclusion includes any nonzero linear combination in a multiple first-positive eigenspace.

The nonvanishing-gradient statement is not a uniform numerical lower bound over the entire domain. The gradient may approach zero as a point approaches the boundary, where extrema are expected. Nor does it assert that one fixed directional derivative is positive throughout the domain.

5. Why the advance matters

Importance: 5/5. If correct, the result settles a central geometric spectral conjecture on the full smooth simply connected planar class and strengthens it by excluding all interior critical points.

The claim would identify a striking constraint on the lowest nonconstant mode: complicated geometry without holes cannot create an interior stationary point. This is much stronger than expecting the hottest point to sit near the boundary in familiar symmetric pictures.

The proposed method also connects two areas that are usually presented separately. A variational principle for PDE vector fields is combined with positivity properties of kernels and Hilbert-space geometry. That connection is part of the family’s mathematical significance, independently of the heat-flow interpretation.

6. The main idea

Use both derivatives together

Each Cartesian derivative of $u$ solves the same Helmholtz equation $(\Delta+\mu_1)v=0$. But neither derivative has a useful fixed sign on an arbitrary domain. The Neumann condition instead says that the vector $\nabla u$ is tangent to the boundary.

Rohleder’s variational principle uses this joint constraint. For tangent vector fields $X=(X_1,X_2)$, define

$$Q(X)=\int_\Omega\bigl(|\operatorname{div}X|^2 +|\operatorname{curl}X|^2\bigr)\,dx,$$

where $\operatorname{curl}X=\partial_1X_2-\partial_2X_1$ in the plane. The principle gives

$$Q(X)\ge\mu_1\int_\Omega|X|^2\,dx,$$

with equality exactly for gradients of functions in $V$.

Simple connectivity removes an extra harmonic-field obstruction in this decomposition. The strict inequality between the first positive Neumann eigenvalue and the first Dirichlet eigenvalue separates the unwanted vector-field component from the equality space.

Convert the problem to boundary data on a circle

A smooth conformal parametrization $\Phi$ maps the unit disk $D$ onto $\Omega$. Pull back the boundary gradient to obtain a vector-valued function $g$ on $S^1$.

Solving the scalar Helmholtz equation in each component produces a nonnegative quadratic form $E$ on tangent boundary data. Its zero-energy vectors correspond precisely to first-eigenfunction gradients; in particular, $E(g)=0$.

Multiplication by any real scalar boundary function $b$ preserves tangency. The manuscript derives the identity

$$E(bg)=\frac12\int_{S^1}\int_{S^1} N(s,t)(b(s)-b(t))^2 \,g(s)\cdot g(t)\,ds\,dt.$$

The kernel $N$ is positive off the diagonal. Nevertheless, the dot product can change sign, so positivity of $N$ alone does not solve the problem.

Choose multipliers through a distance kernel

Let $K(p,s)$ be the kernel evaluating a solution at an interior disk point $p$ from its boundary values. The central analytic claim concerns

$$D_p(s,t)=\frac{K(p,s)K(p,t)}{N(s,t)} \quad(s\ne t), \qquad D_p(s,s)=0.$$

The manuscript proves, as stated, that this is conditionally negative semidefinite: for any finitely many boundary points and real coefficients whose sum is zero,

$$\sum_{i,j}c_ic_jD_p(s_i,s_j)\le0.$$

Schoenberg’s correspondence interprets this as squared distances in a Hilbert space. There is an injective Lipschitz map $b:S^1\to\mathcal H$ such that

$$D_p(s,t)=\|b(s)-b(t)\|_{\mathcal H}^2.$$

Take the scalar coordinates $b_j$ of this map as multipliers. Summing the energy identity cancels the troublesome kernel factor and yields

$$\sum_jE(b_jg)=\frac12|\nabla u(\Phi(p))|^2.$$

This is the desired bridge from an interior critical point to many boundary equality cases.

A critical point would produce too many first modes

If $\Phi(p)$ were critical, the right side would vanish. Every term on the left is nonnegative, so every $b_jg$ must itself be the boundary gradient of another function in $V$.

The manuscript establishes that $V$ has real dimension at most two, using planar separation and the boundary sign pattern of a first eigenfunction. An injective image of a circle cannot lie in a line. When the nonzero set of $g$ is dense, the multipliers therefore produce at least three independent gradient traces, contradicting that dimension bound.

If the nonzero set is not dense, $g$ vanishes on an open boundary arc. Two independent generated eigenfunctions then yield a nonzero combination with both zero value and zero normal derivative on that arc. Boundary uniqueness forces that combination to vanish identically, another contradiction.

Where the kernel theorem enters

The distance-kernel property is the main analytic work. The manuscript regularizes Green functions and studies finite kernel matrices whose entrywise reciprocals have at most one positive eigenvalue. Rank-one updates, positive quadrature and limits transfer that property to the Green resolvent and then to the boundary kernels.

The matrix condition comes from complete Nevanlinna–Pick kernel theory. The argument must handle singular kernels and the order of boundary limits carefully; a boundary Poisson kernel cannot simply be assumed square-integrable. This analytic construction supplies the special multipliers that ordinary sign arguments lack.

7. Scope, limitations, and open questions

The theorem is for smooth bounded simply connected planar domains and the first positive Neumann eigenspace. It does not extend the claim to domains with holes, where counterexamples exist, to arbitrary higher-dimensional domains, or to higher eigenfunctions.

The statement does not classify boundary critical points or prove that extrema are unique. It permits eigenvalue multiplicity and does not provide a domain-independent lower bound for the interior gradient.

The heat-flow interpretation remains asymptotic and depends on which modes are present initially. The theorem does not say that every temperature profile has boundary-only hot spots at every time.

Lower boundary regularity, quantitative gradient estimates and extensions of the reciprocal-kernel method are separate questions beyond the stated theorem.

8. Sources and further reading

The counterexamples, geometric special cases and Rohleder papers linked above supply the historical setting. The triangle result should be read together with its erratum. Optional background on Neumann boundary conditions and conformal maps complements the definitions used here.

The manuscript below supplies the strict hot-spots and noncriticality claims. Its kernel theorem is also formulated independently for bounded nonnegative disk potentials satisfying the specified subcritical Dirichlet inequality.

Source manuscripts

• M1. OpenAI, Strict hot spots and absence of interior critical points on smooth simply connected planar domains (September 24, 2026).

Sources consulted on 26 October 2026 at repository revision 1581d4c. Download the manuscript citations.