The hot spots conjecture can be false: some numerical examples

A Counterexample to the “hot Spots” Conjecture

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.

On the “hot Spots” Conjecture of J. Rauch

The “hot Spots” Conjecture for Nearly Circular Planar Convex Domains

We prove the “hot spots” conjecture of J. Rauch in the case that the domain Ω is a planar convex domain satisfying diam(Ω)2/|Ω| < 1.378. Specifically, we show that an eigenfunction corresponding to the lowest nonzero eigenvalue of the Neumann Laplacian on Ω attains its maximum (minimum) at points on ∂Ω. When Ω is a disk, diam(Ω)2/|Ω| t 1.273. Hence, the above condition indicates that Ω is a nea...

An inequality for potentials and the “hot–spots” conjecture

Consider a nonnegative continuous potential V on the half disk ID = {z = x + iy : y > 0, |z| < 1} for which r2V (reiθ) is nondecreasing as a function of r for every fixed 0 < θ < π. We prove an inequality for the distribution of the random variable ∫ τID+ 0 V (Bs)ds where Bs is the Brownian motion reflected on the top portion of the boundary and killed on the lower portion and τID+ is its lifet...

What Can Seismology Say About Hot Spots?

Seismology offers the highest-resolution view of mantle structure. In the decades since Morgan [1971] first proposed deep-mantle plumes, seismologists have used increasingly sophisticated methods to look for evidence of such structures, but so far they have had little success. This abstract outlines the relevant seismological methods for non-specialists and summarizes the current state of knowl...