Calculus of Variations and Geometric Measure Theory

How to locate the hot spot in a heat conductor?

Lorenzo Brasco (Università degli Studi di Ferrara)

created by magnani on 11 Apr 2011

13 apr 2011


Dipartimento di Matematica - Sala Seminari - ore 17:00

ABSTRACT: One considers the following physical situation: we have a heat conductor ''Omega'' having uniform temperature (say equal to 1, up to normalizations) at the initial time t=0. Then we ground it, i.e. we constantly keep its boundary at zero temperature, and we look at the evolution in time of its temperature, which we suppose to be ruled by the heat equation. What can be said about the location of the hot spots, i.e. the points where for each time the maximal temperature is attained? In this seminar, we illustrate two methods for estimating the location of these points, which are particularly useful when ''Omega'' is a convex set: indeed in this case, one can ensure that at each time we just have a unique hot spot. These methods also apply to the location of some other important points (Santalo` point, maximum point of the first Dirichlet eigenfunction, etc.). These results are contained in a recent work joint with Rolando Magnanini and Paolo Salani (University of Florence).