The location of the hot spot in a grounded convex conductor

TL;DR

This paper studies the precise location of the hot spot in convex heat conductors with grounded boundaries, introducing new mathematical methods and providing numerical results.

Contribution

It introduces two novel mathematical approaches for locating the hot spot in convex conductors and simplifies the problem for polyhedral shapes.

Findings

Methods based on maximum principles and reflection principles effectively locate the hot spot.

Simplification for polyhedral conductors makes the problem more tractable.

Numerical computations support the theoretical findings.

Abstract

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-Amp\`ere 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.