Necessity of connectivity for boundary Dirichlet regions in the mixed hot-spots problem

On a bounded convex planar domain with Dirichlet conditions on a set D contained in the boundary and Neumann conditions elsewhere, Hatcher (J. Spectr. Theory, 2025) proved that the first mixed Laplace eigenfunction has no interior critical points provided D is connected and sufficiently small (diameter below an explicit threshold, or spectrally whenever the first mixed eigenvalue satisfies a bound involving the first Bessel zero j_0 and the domain diameter). The paper itself explicitly leaves one issue open (Remark 1.2): whether the connectivity hypothesis on D is necessary at all when D is contained in the boundary (for Dirichlet regions compactly contained in the domain, connectivity is provably necessary). The open problem is to determine, by counterexample or by theorem, whether connectedness of a boundary-supported Dirichlet region is needed in the smallness regime.

Background

The hot spots conjecture of J. Rauch (posed in 1974) predicts that on a bounded planar domain with insulated (Neumann) boundary, the second Neumann eigenfunction of the Laplacian — which shapes the long-time profile of heat flow from generic initial data — attains its maximum and minimum only on the boundary; Bañuelos and Burdzy (1999) reformulated and popularized the modern version. The conjecture remains open even for general convex planar domains, although it is known for triangles, longish domains, lip domains, and various symmetric classes.

A natural generalization replaces part of the insulated boundary by a held-at-zero-temperature (Dirichlet) region D: one imposes u = 0 on D and the outward normal derivative ∂νu = 0 on the remaining boundary, and asks where the first eigenfunction of this mixed Dirichlet–Neumann problem, u_1^D, attains its maximum. Physically, D is a perfectly refrigerated spot on an otherwise insulated membrane; the maximum of u_1^D is the hot spot of the steady heat profile. Since u_1^D is positive and vanishes on D, its maximum is driven away from D; the question is whether it must lie on the boundary, i.e. whether u_1^D has no critical points in the interior.

Hatcher (2024) proved such results for structured Dirichlet data (e.g. line segments), and the 2024–2025 literature saw a wave of related work (Aldeghi–Rohleder, Li–Yao, and others). Hatcher (J. Spectr. Theory 15 (2025), 1367–1382) proves the small-Dirichlet-region theorem: if Ω is convex, D is connected with diameter at most $\epsilon = \sqrt{|\Omega|/\pi}\,\exp\big(-(4\pi/j_0^2)\, d^2/|\Omega|\big)$ — where d = diam(Ω), |Ω| is the area, and j_0 ≈ 2.40483 is the first positive zero of the Bessel function $J_0$ — then u_1^D has no critical points in Ω \ D. The proof combines a Miyamoto-type comparison argument — comparison of $u_1^D$ with explicitly computable extremal eigenfunctions, in the spirit of Miyamoto's hot-spots theorem for thin convex domains — with new shape-optimization estimates for the first mixed eigenvalue $\lambda_1^D$. More generally, the comparison step (Proposition 1.5) works for Neumann convex pairs (Ω, D), meaning ∂Ω \ D lies on the boundary of the convex hull of Ω \ D: if D is connected and $\lambda_1^D \le (j_0/d)^2$, then u_1^D has no interior critical points. Throughout, D is assumed such that eigenfunctions vanish pointwise on it, e.g. the closure of a relatively open subset of the boundary.

Two limitations are visible in the paper itself. First, the connectivity hypothesis: the paper shows connectivity is necessary when D is compactly contained in Ω — Example 4.1 places two tiny disks symmetrically about the center of a square and, since the first mixed eigenvalue is simple and its eigenfunction positive, symmetry forces the center to be an interior critical point; Example 4.2 builds a configuration with three components whose eigenfunction has an interior local extremum. But for D contained in the boundary ∂Ω, the author states it is unclear whether connectivity is necessary or merely an artifact of the proof (Remark 1.2). Notably, the symmetric constructions that work in the interior place the Dirichlet components around the domain's symmetry center, a mechanism unavailable to small boundary-supported sets. Second, the thresholds are explicitly non-sharp: for the unit disk the theorem's ε equals $\exp(-16/j_0^2) \approx 0.063$, while the paper's own estimates (Theorem 1.6, Example 4.3) show that a single boundary arc of length up to about π/2, of diameter √2, already yields no interior critical points — a gap of more than a factor of twenty. Whether the spectral threshold $\lambda_1^D \le (j_0/d)^2$ is sharp, and what the optimal dependence of the admissible size of D on the geometry of Ω is, is left unresolved in the paper, which documents this gap through its examples without posing it as an open question; the open question the source explicitly poses concerns only the connectivity hypothesis. Subsequent work by the author and others (constant-curvature domains, cones and warped products, hyperbolic domains with small eigenvalues, symmetric quadrangles, curvilinear sectors) extends the surrounding theory but in every case still assumes a connected Dirichlet part, and none addresses the connectivity issue.

Problem Statement

Let Ω ⊂ R² be a bounded convex domain with diameter $d$ and area $|\Omega|$, let $j_0 \approx 2.40483$ denote the first positive zero of the Bessel function $J_0$, and let D ⊆ ∂Ω be the closure of a relatively open subset of the boundary (so that eigenfunctions of the mixed problem vanish pointwise on D). Consider the mixed eigenvalue problem

$$-\Delta u = \lambda u \;\text{ in } \Omega, \qquad u = 0 \;\text{ on } D, \qquad \partial_\nu u = 0 \;\text{ on } \partial\Omega \setminus D,$$

with first eigenvalue $\lambda_1^D$ and first eigenfunction $u_1^D$ (a critical point meaning a point where $\nabla u = 0$). Hatcher proved that if D is connected and $\operatorname{diam}(D) \le \epsilon(\Omega) := \sqrt{|\Omega|/\pi}\,\exp\big(-(4\pi/j_0^2)\, d^2/|\Omega|\big)$, then $u_1^D$ has no critical points in Ω \ D; more generally the same conclusion holds for a Neumann convex pair (Ω, D) — meaning that ∂Ω \ D is contained in the boundary of the convex hull of Ω \ D — with D connected and $\lambda_1^D \le (j_0/d)^2$. The open problem, in the formulation left by the source (Remark 1.2), is the following question on the connectivity of boundary-supported Dirichlet regions. Determine whether the hypothesis that D be connected is necessary when D is contained in ∂Ω. Either (a) exhibit a convex (or Neumann convex) planar domain Ω and a disconnected D ⊆ ∂Ω — the diameter of a disconnected set being the supremum of distances between any two of its points, including points in different components — with diameter at most $\epsilon(\Omega)$ such that $u_1^D$ has a critical point in Ω \ D; or (b) prove that, in the same smallness regime, the absence of interior critical points persists without any connectivity assumption on the boundary-supported Dirichlet region. For Dirichlet regions compactly contained in Ω this is settled: connectivity is necessary there, by the symmetric two- and three-disk examples of the source.

References

1. Lawford Hatcher, "A hot spots theorem for the mixed eigenvalue problem with small Dirichlet region", Journal of Spectral Theory 15 (2025), no. 3, 1367–1382. DOI: 10.4171/jst/569, https://doi.org/10.4171/jst/569; arXiv:2409.03908, https://arxiv.org/abs/2409.03908.
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