Euclidean triangles have no hot spots

Judge, Chris; Mondal, Sugata

doi:10.4007/annals.2020.191.1.3

Cited by 27 publications

(21 citation statements)

References 2 publications

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“…Some partial results can be found in work by Kawohl, Bañuelos-Burdzy, Jerison-Nadirashvili, Atar-Burdzy, Miyamoto, and Siudeja. A proof for acute triangles was announced only a few months ago by Judge and Mondal [18] where references to this earlier work can be found. Despite the resistance of this conjecture, we believe it for simply-connected planar domains and for convex domains in all dimensions.…”

Section: The Hot Spots Conjecture and Lipschitz Level Setsmentioning

confidence: 99%

“…Indeed, this complexity is already present in acute triangles and is a reason why the recent proof of the hot spots conjecture in that case is subtle. For many acute triangles, the least energy non-constant Neumann eigenfunction is not strictly monotone in any direction and has four critical points on the boundary, namely, local maxima and minima at the vertices and a saddle point on one side (see [18]). The rectangle is a borderline symmetric case showing that the failure of convexity can give eigenfunctions that are not monotone and whose level sets are more complicated.…”

Section: The Hot Spots Conjecture and Lipschitz Level Setsmentioning

confidence: 99%

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The Two Hyperplane Conjecture

Jerison

2019

Acta. Math. Sin.-English Ser.

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We introduce a conjecture that we call the Two Hyperplane Conjecture, saying that an isoperimetric surface that divides a convex body in half by volume is trapped between parallel hyperplanes. The conjecture is motivated by an approach we propose to the Hots Spots Conjecture of J. Rauch using deformation and Lipschitz bounds for level sets of eigenfunctions. We will relate this approach to quantitative connectivity properties of level sets of solutions to elliptic variational problems, including isoperimetric inequalities, Poincaré inequalities, Harnack inequalities, and NTA (non-tangentially accessibility). This paper mostly asks questions rather than answering them, while recasting known results in a new light. Its main theme is that the level sets of least energy solutions to scalar variational problems should be as simple as possible.2010 Mathematics Subject Classification. 35B35,35A15.

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Section: The Hot Spots Conjecture and Lipschitz Level Setsmentioning

confidence: 99%

Section: The Hot Spots Conjecture and Lipschitz Level Setsmentioning

confidence: 99%

The Two Hyperplane Conjecture

Jerison

2019

Acta. Math. Sin.-English Ser.

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“…Any bounded domain in R is a finite interval and it is a trivial matter to check that the Hot Spots conjecture holds. As for higher dimensions, besides parallelepipeds, balls, and annuli [Kaw85], some notable examples are triangles [JM20], convex planar domains with two axes of symmetry [JN00], lip domains [AB04], and affine Weyl group alcoves in R d [DM09]; see also [Pas02,BPP04,Siu15]. More generally, HS2 has been shown to hold for bounded cylinders in R d whose cross sections can be arbitrary Lipschitz domains in R d−1 ; see [Kaw85,Corollary 2.15].…”

Section: Introduction 1hot Spots Conjecturementioning

confidence: 99%

Improved upper bounds for the Hot Spots constant of Lipschitz domains

Mariano¹,

Panzo²,

Wang³

2021

Preprint

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The Hot Spots constant for bounded smooth domains was recently introduced by Steinerberger [Ste21] as a means to control the global extrema of the first nontrivial eigenfunction of the Neumann Laplacian by its boundary extrema. We generalize the Hot Spots constant to bounded Lipschitz domains and show that it leads to an if and only if condition for the weak Hot Spots conjecture HS2 of [BB99]. We also derive a new general formula for a dimension-dependent upper bound that can be tailored to any specific class of domains. This formula is then used to compute upper bounds for the Hot Spots constant of the class of all bounded Lipschitz domains in R d for both small d and for asymptotically large d that significantly improve upon the existing results.

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“…. ) the maximum and minimum of the eigenfunction(s) of the first nontrivial eigenvalue of the Neumann Laplacian are on the boundary of the domain; see, e.g., the introduction of [15] for a motivation and an historical description of the problem on domains, [23] for the famous counterexample to the original conjecture, and [38,45,54,55] for recent advances on the problem. The idea behind the conjecture comes from the corresponding heat equation: an expansion of solutions as Fourier series in the eigenfunctions shows that the maximum and minimum of the first nontrivial eigenfunctions represent the generically hottest and coldest points in the domain; and as heat flow should respect the geometry of the domain, it is natural to expect these points to be located far away from each other in some reasonable sense.…”

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confidence: 99%

On the hot spots of quantum graphs

Kennedy¹,

Rohleder²

2021

CPAA

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We undertake a systematic investigation of the maxima and minima of the eigenfunctions associated with the first nontrivial eigenvalue of the Laplacian on a metric graph equipped with standard (continuity-Kirchhoff) vertex conditions. This is inspired by the famous hot spots conjecture for the Laplacian on a Euclidean domain, and the points on the graph where maxima and minima are achieved represent the generically "hottest" and "coldest" spots of the graph. We prove results on both the number and location of the hot spots of a metric graph, and also present a large number of examples, many of which run contrary to what one might naïvely expect. Amongst other results we prove the following: (i) generically, up to arbitrarily small perturbations of the graph, the points where minimum and maximum, respectively, are attained are unique; (ii) the minima and maxima can only be located at the vertices of degree one or inside the doubly connected part of the metric graph; and (iii) for any fixed graph topology, for some choices of edge lengths all minima and maxima will occur only at degree-one vertices, while for others they will only occur in the doubly connected part of the graph.

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Euclidean triangles have no hot spots

Cited by 27 publications

References 2 publications

The Two Hyperplane Conjecture

The Two Hyperplane Conjecture

Improved upper bounds for the Hot Spots constant of Lipschitz domains

On the hot spots of quantum graphs

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