The hot spots conjecture for bounded convex planar domains

The hot spots conjecture, posed by Jeffrey Rauch in 1974, predicts that in a bounded body with perfectly insulating boundary, the long-run hottest and coldest points — the locations of the maximum and minimum of any eigenfunction belonging to the first nonzero Neumann eigenvalue $\mu_2$ of the Laplacian — lie on the boundary. The conjecture is false for some multiply connected planar domains and for convex domains in sufficiently high dimensions, and it has been proved for triangles, symmetric domains, lip domains, and domains close to a disk or to an interval; for general bounded convex domains in the plane it remains open. The problem is to prove, or disprove by an explicit counterexample, that for every bounded convex planar domain every $\mu_2$-eigenfunction attains its global maximum and global minimum only on the boundary.

Background

Consider the heat equation $u_t=\Delta u$ on a bounded connected domain $\Omega$ with insulated boundary, modeled by the Neumann condition $\nabla u\cdot\nu=0$ on $\partial\Omega$, where $\nu$ is the exterior unit normal (zero normal derivative means no heat flux through the boundary). Expanding the initial temperature in eigenfunctions of the Neumann Laplacian, the constant mode (the spatial average) belongs to the eigenvalue $0$, and the first nonzero eigenvalue $\mu_2(\Omega)$ governs the slowest-decaying mode: for large times $t$, the temperature is approximately the average plus $e^{-\mu_2 t}\psi_2(x)$, where $\psi_2$ is an eigenfunction for $\mu_2$ (if $\mu_2$ has multiplicity greater than one, the whole eigenspace enters). Hence the eventual hottest and coldest points of an insulated body are the maximum and minimum points of $\psi_2$ — the “hot spots”. In 1974 Jeffrey Rauch conjectured that for a bounded domain these extrema can occur only on the boundary $\partial\Omega$.

The conjecture is false in unrestricted generality. Burdzy and Werner (1999) constructed a multiply connected planar domain whose second Neumann eigenfunction has an interior extremum, Burdzy (2005) gave further counterexamples for planar domains with one hole, and de Dios Pont (2024) constructed convex counterexamples in $\mathbb{R}^d$ for all sufficiently large $d$, so convexity alone does not force boundary extrema in high dimension. No counterexample is known in the plane for simply connected, in particular convex, domains; the problem here is exactly this surviving planar convex case, which Rohleder’s 2026 note describes as widely believed to be true.

The conjecture has been confirmed only on restricted classes: domains with a line of symmetry (Bañuelos–Burdzy, 1999); domains with two perpendicular axes of symmetry (Jerison–Nadirashvili, 2000); lip domains, i.e. regions trapped between the graphs of two Lipschitz functions (Atar–Burdzy, 2004); all Euclidean triangles (Judge–Mondal, 2020, with a 2022 erratum); planar convex domains sufficiently close to a disk (Miyamoto, 2009); thin curved strips (Krejčiřík–Tušek, 2019); certain non-convex polygons tiled by L-shapes (Hatcher, 2025); and symmetric quadrangles such as isosceles trapezoids, parallelograms, and kites (Deng–Gui–Jiang–Yang–Yao–Zou, 2026). A complementary programme quantifies where the extrema can lie on general convex domains without settling the conjecture: Steinerberger (2020) proved that the maximum and minimum of $\psi_2$ lie within $c\cdot\operatorname{inrad}(\Omega)$ of the two diametral “tips” of $\Omega$, where $\operatorname{inrad}(\Omega)$ is the inradius (the radius of the largest inscribed disk) and $c>0$ is a universal constant; Rohleder (2026) proved that any interior critical point of $\psi_2$ stays out of a central subregion — more precisely, it has distance greater than $(j_1/(2j_0))\operatorname{diam}(\Omega)\approx 0.7967\operatorname{diam}(\Omega)$ from some boundary point, where $j_0$ and $j_1$ are the first positive zeros of the Bessel functions $J_0$ and $J_1$ of orders zero and one, the proof using Kröger’s bound $\mu_2\le 4j_0^2/\operatorname{diam}(\Omega)^2$ — and that $\psi_2$ has no interior critical points at all when $\mu_2(\Omega)\le j_1^2/\operatorname{diam}(\Omega)^2$, a condition satisfied by convex domains close to an interval (via the Payne–Weinberger lower bound $\mu_2\ge\pi^2/\operatorname{diam}(\Omega)^2$, which is approached in the limit as $\Omega$ degenerates to an interval, together with $j_1/\pi\approx 1.2197$). These partial location results leave the middle band of a general convex domain uncontrolled, and no available technique — Brownian-motion coupling, symmetry decomposition, variational principles for gradients of eigenfunctions, or nodal-line analysis — extends to arbitrary convex planar domains. Notably, whether $\mu_2$ is simple for a general convex planar domain is itself open, so both statements and proofs must account for a possibly multi-dimensional eigenspace.

Problem Statement

Let $\Omega\subset\mathbb{R}^2$ be an arbitrary bounded convex open set, and let $\psi_2$ be any real-valued eigenfunction of the Laplacian on $\Omega$ with Neumann boundary condition $\nabla\psi_2\cdot\nu=0$ on $\partial\Omega$, associated with the first nonzero Neumann eigenvalue $\mu_2(\Omega)$ (the eigenspace may have dimension greater than one, and the question covers every member of it). Must $\psi_2$ attain its global maximum and its global minimum only on the boundary $\partial\Omega$?

An affirmative answer for every bounded convex planar domain proves the hot spots conjecture in its surviving planar convex form — the case of Rauch’s 1974 conjecture left open after the multiply connected planar and high-dimensional convex counterexamples. A single explicit bounded convex $\Omega\subset\mathbb{R}^2$ together with a $\mu_2(\Omega)$-eigenfunction attaining a strict global extremum at an interior point refutes it. No additional hypotheses — symmetry, smoothness or strict convexity of the boundary, eccentricity, thinness, or simplicity of $\mu_2(\Omega)$ — may be imposed.

References

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