No interior extrema for first mixed Dirichlet–Neumann eigenfunctions with a single-segment Dirichlet set (Hatcher's Conjecture 1.8), and the L-shaped eigenvalue inequalities of Remark 6.2

This problem records the two crisp open questions left by Hatcher's paper 'The hot spots conjecture for some non-convex polygons' (SIAM Journal on Mathematical Analysis, 2025) on the mixed Dirichlet–Neumann eigenvalue problem for the planar Laplacian. The primary question, Conjecture 1.8, asserts that if Ω is a bounded, simply connected Lipschitz domain in the plane and the Dirichlet set D is a single relatively open line segment (with Neumann conditions on the remaining boundary), then every first mixed eigenfunction has no interior extrema; since such eigenfunctions are positive and strictly superharmonic, this amounts to the absence of interior local maxima, and in particular implies that the maximum is attained only on the Neumann boundary. The secondary question, Remark 6.2, conjectures that two inequalities between the lowest mixed eigenvalues of an L-shaped domain — proved in the paper only under a reflection-symmetry hypothesis — hold for every L-shaped domain. As of August 2026 no proof or counterexample of either conjecture is known: all known positive results are restricted to special geometry classes, and the single-segment conjecture is open even for an L-shaped domain whose segment lies strictly inside an edge. The task is to prove or refute Conjecture 1.8 in its stated generality, and to prove or refute the L-shaped eigenvalue inequalities.

Background

The hot spots conjecture, posed by J. Rauch in 1974, predicts that the first non-constant Neumann eigenfunction of the Laplacian on a bounded Lipschitz domain attains its maximum and minimum only on the boundary. (Neumann conditions, $\partial_\nu u=0$ on the whole boundary, model a perfectly insulated body, and the extrema of the first non-constant eigenfunction describe where heat concentrates in the long-time asymptotics of the heat equation — hence the name hot spots.) The conjecture is open in full generality: it fails on some planar domains with holes (Burdzy and Werner, 1999) and, as recently shown, on convex domains in sufficiently high dimensions (de Dios Pont, 2024), but it is widely believed for contractible planar domains. Hatcher's 2025 paper proves it for L-shaped domains and for five further classes of domains tiled by them, and the same paper is the source of the present open problem.

This problem concerns the mixed Dirichlet–Neumann eigenvalue problem, in which the boundary of a bounded planar domain $\Omega$ is split into a relatively open Dirichlet part $D$, where $u=0$ (an absorbing or 'cold' portion of the boundary), and a Neumann part $N=\partial\Omega\setminus\overline{D}$, where the normal derivative $\partial_\nu u=0$ (an insulating portion): $-\Delta u=\lambda u\ \text{in}\ \Omega,\qquad u=0\ \text{on}\ D,\qquad \partial_\nu u=0\ \text{on}\ N.$ When $D$ has positive boundary measure, the lowest eigenvalue $\lambda_1^D$ is positive, its eigenfunctions do not change sign (by Courant's nodal domain theorem, which bounds the number of nodal regions of a low eigenfunction), and one takes $u>0$ in $\Omega$; under the piecewise-smooth hypotheses used in the literature the eigenvalue is in addition simple. Because $\Delta u=-\lambda_1^D u<0$, the eigenfunction is strictly superharmonic and real-analytic inside $\Omega$, so an interior local minimum cannot occur (the Laplacian is non-negative at a local minimum) and the infimum $0$ is attained on $D$. Thus 'no interior extrema' is exactly the statement that no interior local maximum exists. The natural mixed analogue of the hot spots question, raised by Banuelos–Pang and Banuelos–Pang–Pascu, asks: for which triples $(\Omega,D,N)$ does the first mixed eigenfunction attain its maximum only on the boundary? The naive analogue is false even on simply connected planar domains: for a rectangle with $D$ the union of two opposite edges and $N$ the union of the other two, the first eigenfunctions are of the type $(x,y)\mapsto\sin(\pi x/w)$ and attain their maximum along an interior segment. So hypotheses on $D$ are genuinely needed.

Hatcher's Conjecture 1.8 asserts that a single segment always suffices: if $\Omega$ is a bounded, simply connected Lipschitz domain in the plane and $D$ is a single relatively open line segment, then each first mixed eigenfunction has no interior extrema. The currently known positive results are all restricted to special classes: convex planar domains bounded by two curves meeting at angles smaller than $\pi/2$, one of which may be a line segment (Banuelos–Pang; Banuelos–Pang–Pascu); simply connected Lipschitz domains whose boundary normals satisfy quadrant conditions, with acute Dirichlet–Neumann junctions and no inward corners (Aldeghi–Rohleder); triangles, where the first mixed eigenfunction has at most one critical point and it lies on the Neumann boundary (Hatcher); graph domains, i.e. regions between the x-axis and the graph of a positive piecewise-smooth function, with the Neumann part equal to the bottom edge (Hatcher); convex domains with $D$ connected and sufficiently small (Hatcher); rectangles with $D$ an arbitrary segment contained in an edge, where the critical points are confined to the opposite edge (Hatcher); and L-shaped domains when $D$ is a full edge or a union of edges under side conditions (Hatcher). The conjecture remains open even for an L-shaped domain when $D$ is a proper sub-segment of an edge, since the L-shaped theorem requires full edges.

An L-shaped domain is the open hexagon with vertices $(0,0)$, $(a_1+a_2,0)$, $(a_1+a_2,a_3)$, $(a_1,a_3)$, $(a_1,a_3+a_4)$, $(0,a_3+a_4)$ with all $a_i>0$; the vertex $(a_1,a_3)$ is the re-entrant (non-convex) corner, near which eigenfunctions behave like $r^{2/3}$ rather than being smooth. Its edges are labelled $e_1,\dots,e_6$ counterclockwise, with $e_1$ the edge contained in the x-axis; for a union of edges $E\subseteq\partial L$, denote by $\lambda_1^E$ the lowest mixed eigenvalue with Dirichlet set $E$. Hatcher's Lemma 6.1 proves, under the reflection symmetry $a_1=a_3$ and $a_2=a_4$, that $\lambda_1^{e_1}<\lambda_1^{e_3\cup e_5}$ and $\lambda_1^{e_2}<\lambda_1^{e_4\cup e_6}$, extending mixed eigenvalue inequalities of Lotoreichik–Rohleder. These orderings are the comparison inputs behind the paper's theorems locating the critical points of mixed eigenfunctions on L-shaped and L-tiled domains; Remark 6.2 records the conjecture, supported by numerical experimentation, that the same inequalities hold for every L-shaped domain, together with the statement that the author does not know how to prove them in full generality.

Problem Statement

Primary question (Hatcher's Conjecture 1.8, in the bounded planar setting of the source paper). Prove or refute the following: for every bounded, simply connected Lipschitz domain $\Omega\subset\mathbb{R}^2$ and every relatively open line segment $D\subset\partial\Omega$ (that is, a line segment that is open in the topology of $\partial\Omega$), with $N=\partial\Omega\setminus\overline{D}$, each eigenfunction corresponding to the lowest eigenvalue $\lambda_1^D$ of $-\Delta u=\lambda u\ \text{in}\ \Omega,\qquad u=0\ \text{on}\ D,\qquad \partial_\nu u=0\ \text{on}\ N$ has no interior extrema; equivalently, since such eigenfunctions are positive and strictly superharmonic (so no interior local minimum can occur), no first mixed eigenfunction has an interior local maximum, and hence in particular the maximum of every first mixed eigenfunction is attained only on the Neumann part of the boundary. A resolution either proves the statement for all such pairs $(\Omega,D)$, or exhibits one admissible pair $(\Omega,D)$ together with a certified first mixed eigenfunction possessing an interior local extremum.

Secondary question (Remark 6.2 of the same paper). Prove or refute: for every L-shaped domain $L(a_1,a_2,a_3,a_4)$ with all $a_i>0$, with edges labelled $e_1,\dots,e_6$ counterclockwise and $e_1$ contained in the x-axis, $\lambda_1^{e_1}<\lambda_1^{e_3\cup e_5}\qquad\text{and}\qquad \lambda_1^{e_2}<\lambda_1^{e_4\cup e_6},$ where $\lambda_1^E$ denotes the lowest mixed eigenvalue with Dirichlet set $E$. A resolution either proves both inequalities for all parameter values, or exhibits an L-shaped domain with a certified violation of at least one of them.

These two questions form one coherent objective: completing the mixed hot-spots analysis that Hatcher's paper initiates. The primary conjecture is the qualitative target, and the secondary inequalities are the eigenvalue-ordering facts that would extend the paper's critical-point theorems from reflection-symmetric L-shaped domains to all of them.

References

1. Lawford Hatcher, 'The hot spots conjecture for some non-convex polygons', SIAM Journal on Mathematical Analysis (2025), DOI 10.1137/24m170702x; preprint arXiv:2405.19508, https://arxiv.org/abs/2405.19508 (source of Conjecture 1.8 and Remark 6.2)
2. Lawford Hatcher, 'First mixed Laplace eigenfunctions with no hot spots', Proceedings of the American Mathematical Society (2024), DOI 10.1090/proc/16939; preprint arXiv:2401.01514, https://arxiv.org/abs/2401.01514
3. Lawford Hatcher, 'A hot spots theorem for the mixed eigenvalue problem with small Dirichlet region', Journal of Spectral Theory (2025), DOI 10.4171/jst/569; preprint arXiv:2409.03908, https://arxiv.org/abs/2409.03908
4. Nausica Aldeghi and Jonathan Rohleder, 'On the first eigenvalue and eigenfunction of the Laplacian with mixed boundary conditions', Journal of Differential Equations (2025), DOI 10.1016/j.jde.2025.02.006; preprint arXiv:2403.17717, https://arxiv.org/abs/2403.17717
5. Rui Li and Ruofei Yao, 'Monotonicity of positive solutions to semilinear elliptic equations with mixed boundary conditions in triangles', Journal of Functional Analysis (2026), DOI 10.1016/j.jfa.2026.111448; preprint arXiv:2401.17912, https://arxiv.org/abs/2401.17912
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9. Rodrigo Banuelos and Michael Pang, 'An inequality for potentials and the hot-spots conjecture', Indiana University Mathematics Journal 53 (2004), DOI 10.1512/iumj.2004.53.2340, https://doi.org/10.1512/iumj.2004.53.2340
10. Vladimir Lotoreichik and Jonathan Rohleder, 'Eigenvalue inequalities for the Laplacian with mixed boundary conditions', Journal of Differential Equations (2017), DOI 10.1016/j.jde.2017.02.043, https://doi.org/10.1016/j.jde.2017.02.043
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12. Krzysztof Burdzy and Wendelin Werner, 'A counterexample to the hot spots conjecture', Annals of Mathematics 149 (1999), DOI 10.2307/121027, https://doi.org/10.2307/121027
13. Jaume de Dios Pont, 'Convex sets can have interior hot spots', preprint arXiv:2412.06344 (2024), https://arxiv.org/abs/2412.06344