Now showing items 1-5 of 5
The "hot spots" problem in planar domains with one hole.
(Duke University Press, 2005)
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.
An annihilating-branching particle model for the heat equation with average temperature zero
We consider two species of particles performing random walks in a domain in [Real numbers] [superscript] d with reflecting boundary conditions, which annihilate on contact. In addition there is a conservation law so that the total number of particles of each type is preserved: When the two particles of different species ...
Fiber Brownian motion and the "hot spots" problem
(Duke University Press, 2000-10)
We show that in some planar domains both extrema of the second Neumann eigenfunction lie strictly inside the domain. The main technical innovation is the use of "fiber Brownian motion," a process which switches between two-dimensional and one-dimensional evolution.
On Neumann eigenfunctions in lip domains
(American Mathematical Society, 2004)
A "lip domain" is a planar set lying between graphs of two Lipschitz functions with constant 1. We show that the second Neumann eigenvalue is simple in every lip domain except the square. The corresponding eigenfunction attains its maximum and minimum at the boundary points at the extreme left and right. This settles the ...
Neumann eigenfunctions and Brownian couplings
(Mathematical Society of Japan, 2004)
This is a review of research on geometric properties of Neumann eigenfunctions related to the "hot spots" conjecture of Jeff Rauch. The paper also presents, in an informal way, some probabilistic techniques used in the proofs.