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The “hot spots” conjecture for domains with two axes of symmetry. (English) Zbl 0948.35029

Consider a convex planar domain \(\Omega \) having a Lipschitz boundary with two axes of symmetry \(x_{1}=0\) and \(x_{2}=0\). The authors show that the extremum of a Neumann eigenfunction with lowest nonzero eigenvalue occurs only at points on the boundary. This result proves J. Rauch’s “hot spots” conjecture: if the initial temperature distribution is not orthogonal to the first nonzero eigenspace, then the point at which the temperature achieves its maximum tends to the boundary. Here, the temperature satisfies the heat equation and its normal derivative vanishes at the boundary. This long paper contains many other results and conjectures. One can imagine the discussed questions by the following two results. 1. Let \(\lambda _{i}\) \((i=1,2)\) are the corresponding lowest Neumann eigenvalues for functions \(\varphi _{i}\) odd in \(\Omega _{i}\) with respect to \(x_{2}=0\), then \(|\lambda _{1}-\lambda _{2}|^{2}\leq c\|\varphi _{1}-\varphi _{2}\|_{L^{\infty }}\). 2. Let \(h(x,t)\) be any solution to the heat equation with zero Neumann boundary conditions and such that \(\int_{\Omega }h(x,0)u(x) dx\neq 0\) for some eigenfunction \(u\) with lowest nonzero eigenvalue. Let \(h(x,t)\) achieves the maximum on \(x\) at the point \(x_{t}\). Then \(x_{t}\) tends to \(\partial \Omega \) as \(t\rightarrow \infty \). If \(\partial \Omega \) is smooth and positively curved, then there is \(T\) such that \(x_{t}\in \partial \Omega \) for \(t\geq T\).

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35B65 Smoothness and regularity of solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
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