Calculus of Variations and Geometric Measure Theory

L. Brasco - R. Magnanini - P. Salani

The location of the hot spot in a grounded convex conductor

created by brasco on 21 Dec 2010
modified on 19 Sep 2011


Accepted Paper

Inserted: 21 dec 2010
Last Updated: 19 sep 2011

Journal: Indiana Univ. Math. J.
Pages: 20
Year: 2010


We investigate the location of the (unique) hot spot in a convex heat conductor with unitary initial temperature and with boundary grounded at zero temperature. We present two methods to locate the hot spot: the former is based on ideas related to the Alexandrov-Bakelmann-Pucci maximum principle and Monge-Ampère equations; the latter relies on Alexandrov's reflection principle. We then show how such a problem can be simplified in case the conductor is a polyhedron. Finally, we present some numerical computations.

Keywords: Heat equation, hot spot, eigenfunctions, Santalò point