Hot spots in convex domains are in the tips (up to an inradius)

Abstract Let be a bounded, convex domain and let 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 satisfy then every maximum and minimum is assumed within distance of x1 and x2, where c is a universal constant (which is the optimal scaling up to the value of c).

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