相关论文: Location of hot spots in thin curved strips

The second eigenfunction of the Neumann Laplacian on convex, planar domains is considered. Inspired by the famous hot spots conjecture and a related result of Steinerberger, we show that potential critical points of this eigenfunction (and,…

Let $\Omega \subset \mathbb{R}^2$ be a bounded, convex domain and let $-\Delta \phi_1 = \mu_1 \phi_1$ be the first nontrivial Laplacian eigenfunction with Neumann boundary conditions. The Hot Spots conjecture claims that the maximum and…

We give an elementary new proof of the hot spots conjecture for L-shaped domains. This result, in addition to a new eigenvalue inequality, allows us to locate the hot spots in Swiss cross translation surfaces. We then prove, in several…

We prove a variant of Rauch's hot spots conjecture for hyperbolic planar domains with small Neumann or mixed Dirichlet-Neumann eigenvalues. We conclude, for instance, that on bounded convex domains in the hyperbolic plane with sufficiently…

The hot spots conjecture asserts that for any convex bounded domain $\Omega$ in $\mathbb R^d$, the first non-trivial Neumann eigenfunction of the Laplace operator in $\Omega$ attains its maximum at the boundary. We construct counterexamples…

The hot spots conjecture of J. Rauch states that the second Neumann eigenfunction of the Laplace operator on a bounded Lipschitz domain in $\mathbb{R}^n$ attains its extrema only on the boundary of the domain. We present an analogous…

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…

We prove constant-curvature analogues of several results regarding the hot spots conjecture in dimension two. Our main theorem shows that the hot spots conjecture holds for all non-acute geodesic triangles of constant negative curvature. We…

We show that the hot spots conjecture of J. Rauch holds for acute triangles if one of the angles is not larger than $\pi/6$. More precisely, we show that the second Neumann eigenfunction on those acute triangles has no maximum or minimum…

We study the hot spots conjecture for domains in the Gaussian space $(\mathbb{R}^n, (2\pi)^{-n/2} e^{-|x|^2/2} dx)$ for $n \ge 2$. Given a bounded domain $\Omega$ with a piecewise smooth boundary, we consider the first nontrivial…

We introduce a new variational principle for the study of eigenvalues and eigenfunctions of the Laplacians with Neumann and Dirichlet boundary conditions on planar domains. In contrast to the classical variational principles, its minimizers…

The hot spots conjecture is only known to be true for special geometries. It can be shown numerically that the hot spots conjecture can fail to be true for easy to construct bounded domains with one hole. The underlying eigenvalue problem…

There exists a planar domain with piecewise smooth boundary and one hole such that the second eigenfunction for the Laplacian with Neumann boundary conditions attains its maximum and minimum inside the domain.

We prove the Hot Spot conjecture on the Vicsek set. Specifically, we show that every eigenfunction of the second smallest eigenvalue of the Neumann Laplacian on the Vicsek set attains its maximum and minimum on the boundary.

We construct a counterexample to the ``hot spots'' conjecture; there exists a bounded connected planar domain (with two holes) such that the second eigenvalue of the Laplacian in that domain with Neumann boundary conditions is simple and…

We build a one-parameter family of S^{1}-invariant metrics on the unit disc with fixed total area for which the second eigenvalue of the Laplace operator in the case of both Neumann and Dirichlet boundary conditions is simple and has an…

We investigate the "hot--spots" property for the survival time probability of Brownian motion with killing and reflection in planar convex domains whose boundary consists of two curves, one of which is an arc of a circle, intersecting at…

We review a recent new approach to the study of critical points of Laplacian eigenfunctions. Its core novelty is a non-standard variational principle for the eigenvalues of the Laplacians with Neumann and Dirichlet boundary conditions on…

Let $D \subset \mathbb{R}^d$ be a bounded, connected domain with smooth boundary and let $-\Delta u = \mu_1 u$ be the first nontrivial eigenfunction of the Laplace operator with Neumann boundary conditions. We prove $$ \max_{x \in D} ~u(x)…

The Hot Spots constant for bounded smooth domains was recently introduced by Steinerberger (2021) as a means to control the global extrema of the first nontrivial eigenfunction of the Neumann Laplacian by its boundary extrema. We generalize…