The strong hot spots conjecture in higher dimensions beyond lip and symmetric domains, and the spectrum of the associated vector-valued Laplacian

Rauch's hot spots conjecture predicts that on a bounded insulated domain the eigenfunction of the first nonzero Neumann eigenvalue attains its hottest and coldest points on the boundary; the strong form requires the extrema to be attained only on the boundary. Kennedy and Rohleder proved the strong form for multidimensional lip domains, and for the distinguished eigenfunctions on domains symmetric in all coordinate hyperplanes whose intersection with an orthant is a lip domain, using a purely variational, vector-valued Laplacian whose spectrum encodes the Neumann and mixed Dirichlet-Neumann spectra. This record poses the problem left open by that work: establish the strong hot spots property for explicit classes of bounded piecewise-smooth domains in $\mathbb{R}^d$, $d\ge3$, strictly beyond those two classes — with the flagship instance being bounded convex domains in $\mathbb{R}^3$ and other low dimensions — or refute it for a natural candidate class by an explicit counterexample. Related open questions concern the spectral theory of the framework itself: the characterization of the divergence-free sector $A_2$ in dimensions $d\ge4$, the comparison $\tau_1(C)>\mu_2$ between the first curl-curl and first nonzero Neumann eigenvalues beyond strictly convex $C^{1,1}$ and lip domains, and the minimal boundary regularity; progress on these is partial progress on the framework side, not an answer to the main question. A precise nonempty open core remains: convex counterexamples exist only for all sufficiently large dimensions (de Dios Pont, 2024), and the planar convex case is still open.

Background

Consider the heat equation on a bounded domain $\Omega\subset\mathbb{R}^d$ with a perfectly insulated boundary, i.e. the homogeneous Neumann condition $\partial_\nu u=0$ on $\partial\Omega$, where $\nu$ is the outer unit normal. The Neumann Laplacian $-\Delta_N$ has eigenvalues $0=\mu_1<\mu_2\le\mu_3\le\dots$; after the constants, the first nontrivial eigenfunction $\psi_2$ (with eigenvalue $\mu_2$) governs the long-time behavior of solutions. The physical intuition that the hottest and coldest spots of an insulated body migrate to its surface led Rauch in the 1970s to the hot spots conjecture: $\psi_2$ attains its maximum and minimum on $\partial\Omega$. The strong form asks that these extrema be attained only on $\partial\Omega$. The conjecture is false for completely general domains: Burdzy and Werner (1999) constructed planar domains with holes whose second Neumann eigenfunctions have interior extrema, so research targets structured classes. In the plane the conjecture is known for long and thin domains (R. Bañuelos and K. Burdzy; R. Atar and K. Burdzy), lip domains (Atar–Burdzy: domains enclosed between two Lipschitz graphs with Lipschitz constant at most one, proved via couplings of reflected Brownian motion, i.e. the probabilistic representation of the Neumann heat flow), domains with two axes of symmetry (Jerison–Nadirashvili), convex domains with one axis of symmetry (Pascu), and Euclidean triangles (Judge–Mondal); Steinerberger showed that in convex planar domains hot spots lie within one inradius of a 'tip' of the domain. The case of general convex planar domains remains open. In dimensions $d\ge3$ positive results are far sparser: besides a classical observation for cylindrical domains, the main ones are the lip-type domains of Yang (via reflected Brownian motion) and the deformation-based results of Chen–Li–Wang for certain symmetric domains. Kennedy and Rohleder introduced a purely variational approach: a self-adjoint vector-valued Laplacian $A$ acting on vector fields on $\Omega$, whose quadratic form contains a boundary term involving the shape operator of $\partial\Omega$ (the matrix of normal derivatives of $\nu$, i.e. the curvature of the boundary), and whose spectrum contains the nonzero Neumann and mixed Dirichlet–Neumann eigenvalues; the eigenfunction for the lowest eigenvalue is essentially $\nabla\psi_2$, so sign properties of this vector field (coordinate-wise monotonicity of $\psi_2$) yield the strong hot spots conclusion. The underlying function space splits into a gradient part and a divergence-free part, and $A$ correspondingly splits; the divergence-free sector $A_2$ is well understood only in low dimensions: for simply connected planar $\Omega$ the spectrum of $A_2$ coincides with the Dirichlet spectrum, while in $\mathbb{R}^3$, $A_2$ equals the curl-curl operator $Cu=\operatorname{curl}\operatorname{curl}u$ (the Maxwell eigenvalue operator with the boundary condition $u\times\nu=0$ on tangential fields with $\operatorname{div}u=0$). Kennedy and Rohleder proved the strong form for bounded lip domains in $\mathbb{R}^d$ (after rotation, the outer normal has exactly two nonzero components of opposite sign, or equals $\pm e_j$; together with piecewise $C^\infty$ faces and a uniform exterior ball condition, meaning a ball of fixed radius rolls freely outside $\Omega$), and a companion result for domains symmetric with respect to every coordinate hyperplane whose intersection with some orthant is a lip domain, for the distinguished eigenfunctions that are antisymmetric in one and symmetric in the remaining hyperplanes. Their method also yields $\tau_1(C)>\mu_2$ for lip domains in $\mathbb{R}^3$, where $\tau_1(C)$ is the first curl-curl eigenvalue; Zhang had proved the same comparison for strictly convex $C^{1,1}$ domains. Very recently de Dios Pont constructed smooth, centrally symmetric convex domains in $\mathbb{R}^d$, for all sufficiently large $d$, whose second Neumann eigenfunctions attain interior extrema, refuting the conjecture for convex sets in high dimensions; the higher-dimensional question therefore survives only for structured classes and for low dimensions such as $d=3$.

Problem Statement

Let $\Omega\subset\mathbb{R}^d$, $d\ge3$, be a bounded, connected, piecewise smooth Lipschitz domain (boundary a finite union of $C^\infty$ faces, satisfying a uniform exterior ball condition — a ball of some fixed positive radius can be placed outside $\Omega$ touching each boundary point — as in the source's Hypothesis 1.2). Say that $\Omega$ has the strong hot spots property if every eigenfunction $\psi_2$ of the Neumann Laplacian for the first nonzero eigenvalue $\mu_2$ attains its maximum and its minimum only on $\partial\Omega$. The common objective is to determine how far the strong hot spots conjecture — and the variational vector-operator framework that currently proves it — extends among higher-dimensional domains, beyond the classes settled by Kennedy and Rohleder: multidimensional lip domains (settled in full), and domains symmetric in every coordinate hyperplane whose intersection with some orthant is a lip domain (settled for the eigenfunctions that are antisymmetric in one coordinate hyperplane and symmetric in the remaining ones). Main question. (1) Prove: establish the strong hot spots property for an explicit, natural class of such domains that is not contained in the union of the two settled classes and is genuinely open — the property must not already follow from established results for previously settled families (lip domains, the settled orthant-symmetric class, planar domains with two axes of symmetry, Euclidean triangles, long-and-thin domains), nor from cylindrical products of such families; the flagship open instance is the class of bounded convex domains in $\mathbb{R}^3$ that satisfy the domain regularity above, and establishing the property for this flagship class alone resolves this part; more generally, deciding for each fixed dimension $d$ below the unknown threshold beyond which de Dios Pont's counterexamples apply whether all bounded convex domains in $\mathbb{R}^d$ that satisfy the domain regularity above have the property is a strictly stronger statement whose resolution also settles this part. (2) Refute: exhibit an explicit domain belonging to a natural candidate class not covered by known counterexamples, together with an eigenfunction for $\mu_2$ that attains a strict extremum at an interior point. Related open questions on the spectral side of the variational framework are: a characterization of the divergence-free sector $A_2$ of the vector-valued operator $A$ in dimensions $d\ge4$ — in $d=3$ it is the curl-curl operator $C$ — and of its spectral relation to the Dirichlet and Neumann Laplacians; an extension of the comparison $\tau_1(C)>\mu_2$ from strictly convex $C^{1,1}$ domains and lip domains in $\mathbb{R}^3$ to broader geometric classes and to higher dimensions, with sharp conditions for strictness; and the minimal regularity of the boundary faces — currently piecewise $C^\infty$, weakenable to $C^3$ — under which the conclusions remain valid. These are independent questions: progress on them constitutes partial progress on the framework side, and none of them is a prerequisite for, or a substitute for, an answer to the main question — indeed the source's all-dimensional variational proof (including for lip domains in $d\ge4$) uses no further properties of $A_2$.

References

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