Krzysztof Burdzy
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.
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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 acute angles. This leads to the "hot--spots" property for the mixed Dirichlet--Neumann eigenvalue problem in the domain with Neumann conditions on one of the curves and Dirichlet conditions on the other
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 such that the corresponding eigenfunction attains its strict maximum at an interior point of that domain.
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 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 from Bañuelos and Burdzy (1999). 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 fo…
We investigate the large time behavior of the hot spots of the solution to the Cauchy problem for the heat equation with a potential $\partial_t u-Δu+V(|x|)u=0$, where $V=V(r)$ decays quadratically as $r\to\infty$. In this paper, based on the arguments in [K. Ishige and A. Mukai, preprint (arXiv:1709.00809)], we classify the large time behavior of the hot spots of $u$ and reveal the relationship between the behavior of the hot spots and the harmonic functions for $-Δ+V$.
We show that a second Neumann eigenfunction u of a Euclidean triangle has no critical points in the interior of the triangle.
We investigate the location of the (unique) hot spot in a convex heat conductor with unitary initial temperature and with boundary grounded at zero temperature. We present two methods to locate the hot spot: the former is based on ideas related to the Alexandrov-Bakelmann-Pucci maximum principle and Monge-Ampère equations; the latter relies on Alexandrov's reflection principle. We then show how such a problem can be simplified in case the conductor is a polyhedron. Finally, we present some numerical computations.
Hot spots in convex hyperbolic planar domains with small eigenvalues
本文证明了在具有小Neumann或混合Dirichlet-Neumann特征值的双曲平面域中，Rauch热点猜想的一个变种，得出结论：在双曲平面上足够大的面积有界凸域中，第二Neumann拉普拉斯特征函数没有内部临界点。
The maxima and minima of Neumann eigenfunctions of thin tubular neighbourhoods of curves on surfaces are located in terms of the maxima and minima of Neumann eigenfunctions of the underlying curves. In particular, the hot spots conjecture for a new large class of domains (possibly non-convex and non-Euclidean) is proved.
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 cases, that first mixed Dirichlet-Neumann eigenfunctions of the Laplacian on L-shaped domains also have no interior critical points. As a combination of these results, we prove the hot spots conjecture for five classes of domains tiled by L-shaped domains, including a class of non-simply connected domains. An interesting feature of the proofs is that we make pos…
First mixed Laplace eigenfunctions with no hot spots
本文针对具有混合Dirichlet-Neumann边界条件的区域提出了类似于热点猜想的问题，并证明了欧几里得三角形及特定分段光滑函数图像为边界的平面区域满足该性质。
A note on hot-spots free subregions of convex domains
本文研究了凸域中Neumann拉普拉斯算子第二个特征函数的临界点分布，证明了热点不可能出现在域中心附近，并明确描述了排除区域。
Let $Ω\subset \mathbb{R}^2$ be a bounded, convex domain and let $-Δϕ_1 = μ_1 ϕ_1$ be the first nontrivial Laplacian eigenfunction with Neumann boundary conditions. The Hot Spots conjecture claims that the maximum and minimum are attained at the boundary. We show that they are attained far away from one another: if $x_1, x_2 \in Ω$ satisfy $\|x_1 - x_2\| = \mbox{diam}(Ω)$, then every maximum and minimum is assumed within distance $c\cdot \mbox{inrad}(Ω)$ of $x_1$ and $x_2$, where $c$ is a universal constant (which is the optimal scaling up to the value of $c$).
We investigate the shape of the solution of the Cauchy problem for the damped wave equation. In particular, we study the existence, location and number of spatial maximizers of the solution. Studying the shape of the solution of the damped wave equation, we prepare a decomposed form of the solution into the heat part and the wave part. Moreover, as its another application, we give $L^p$-$L^q$ estimates of the solution.
Convex sets can have interior hot spots
本研究针对热点猜想（凸域上拉普拉斯算子的首个非平凡诺伊曼本征函数最大值在边界），通过将猜想从凸集推广到对数凹测度的方法，构造了所有足够大维度$d$下的反例，证明凸集可存在内部热点。
Removing Boundary Smoothness from the Hot-Spots Conclusion in the Simple-Eigenvalue Case
本文将光滑域的第一正Neumann特征函数内部临界点排除定理归约到满足特定条件的平面域的非严格热点结论，相关论证由ChatGPT基于作者提示生成。
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 for the Laplace equation with Neumann boundary condition is solved with boundary integral equations yielding a non-linear eigenvalue problem. Its discretization via the boundary element collocation method in combination with the algorithm by Beyn yields highly accurate results both for the first non-zero eigenvalue and its corresponding eigenfunction which is…
Hot spots on cones and warped product manifolds
研究了扭曲积流形上热方程热点的极值问题及长期行为，证明了Rauch猜想在特定条件下成立，并揭示了谱间隙对长期行为的影响。
We show that the hot spots conjecture of J. Rauch holds for acute triangles if one of the angles is not larger than $π/6$. More precisely, we show that the second Neumann eigenfunction on those acute triangles has no maximum or minimum inside the domain. We first simplify the problem by showing that absence of critical points on two sides implies no critical points inside a triangle. This result applies to any acute triangle and might help prove the conjecture for arbitrary acute triangles. Then we show that there are no critical points on two sides assuming one small angle. We also establish …
Let $D \subset \mathbb{R}^d$ be a bounded, connected domain with smooth boundary and let $-Δu = μ_1 u$ be the first nontrivial eigenfunction of the Laplace operator with Neumann boundary conditions. We prove $$ \max_{x \in D} ~u(x) \leq 60 \cdot \max_{x \in \partial D} ~u(x)$$ and emphasize that this constant is uniform among all connected domains with smooth boundary in all dimensions. In particular, the Hot Spots Conjecture cannot fail by an arbitrary factor. The inequality also holds for other (Neumann-)eigenfunctions (possibly with a different constant) provided the eigenvalue is smaller t…
A variational approach to the hot spots conjecture
本文回顾了一种基于非标准变分原理研究拉普拉斯特征函数临界点的新方法，并用其为热点猜想的若干已知结果提供简洁证明。
Hot spots in domains of constant curvature
本文在常曲率二维区域中研究热点猜想，证明该猜想对所有非锐角常负曲率测地三角形成立，并进一步刻画了混合 Dirichlet-Neumann 特征函数与第二 Neumann 特征函数的临界点性质。
A hot spots theorem for the mixed eigenvalue problem with small Dirichlet region
本文证明了在凸区域上，当连通的 Dirichlet 区域足够小时，第一混合 Laplace 特征函数无内部临界点，并给出了第一混合特征值的两个新估计及显式示例。
Sharp bounds on the failure of the hot spots conjecture
本文在任意维度下确定了连通 Lipschitz 区域热点比率的最大可能值，证明其随维度趋于无穷时收敛到 $\sqrt{e}$，并由此推出热点猜想在测度意义下渐近成立。
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 are gradients of eigenfunctions instead of the eigenfunctions themselves. This variational principle enables us to give an elementary analytic proof of the famous hot spots conjecture for the class of so-called lip domains. More specifically, we show that each eigenfunction corresponding to the lowest positive eigenvalue of the Neumann Laplacian on such a d…
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 eigenfunction with a closed nodal line. In the case of Neumann boundary conditions, we also prove that this eigenfunction attains its maximum at an interior point, and thus provide a counterexample to the hot spots conjecture on a simply connected surface. This is a consequence of the stronger result that within this family of metrics any given (finite) number …
We use probabilistic tools based on Brownian motion and Feynman-Kac formulae to investigate the heat profile for the ground state Dirichlet and second Neumann eigenfunctions. Among other topics, we comment on supremum norm bounds for ground state Dirichlet eigenfunctions and look at the corresponding Neumann problem, namely the comparison of maximum temperatures on the interior and the boundary, the latter being partially motivated by the hot spots problem. We also investigate the proximity/distance of level sets of ground state Dirichlet eigenfunctions, some with analogous statements for Neum…
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