Analytic Properties of Heat Equation Solutions and Reachable Sets
Speaker: Prof. Dr. Alden Waters
Affiliation: University of Groningen, Netherlands
Abstract: We consider heat equations on bounded Lipschitz domains Omega in R^d and show that solutions to the heat equation for positive times are analytically extendable to a subdomain of the complex plane containing Omega. Our analysis is based on the boundary layer potential method for the heat equation. In particular, our method gives an explanation for the shapes appearing in the literature in 1d, which is not so easy to explain using Fourier analysis alone. I will also discuss the converse theorem, namely that certain sets in the complex plane can be realized as solutions to the heat equation on the boundary of Omega when Omega is a ball. Boundary layer potential theory also gives an indication that this statement is more difficult if Omega is not a ball. This exciting new technique to analyze the question of reachable sets is joint work with Alexander Strohmaier.