Sam Farrington

Heat Flow and Shape Optimisation

In this talk we will look at some shape optimisation problems motivated by the phenomenon of heat flow. In particular, we will mainly focus on the problem of maximising the heat content of a given domain at a given time among various classes of domains and various boundary conditions. Such problems have a strong interplay with other shape optimisation problems, including Faber-Krahn type inequalities, Saint-Venant type inequalities and other isoperimetric inequalities.

We will review known results for maximising the Dirichlet and open heat contents, consider some shape optimisation problems for mixed Dirichlet-Neumann heat content and finally, time-permitting, discuss some ideas around maximising the Robin heat content. Along the way, we will consider some of the probabilistic interpretations of the ideas used and results.