The Matzoh Ball Soup Problem: a complete characterization

TL;DR

This paper fully characterizes solutions to the heat equation with spatial surfaces that do not change over time, providing a complete answer to the Matzoh Ball Soup Problem and extending results to related nonlinear PDEs.

Contribution

It offers a complete classification of solutions with time-invariant spatial equipotential surfaces for the heat equation and related nonlinear PDEs, resolving a longstanding problem.

Findings

Solutions are either isoparametric or split in space-time.

Provides a complete characterization of solutions with invariant equipotential surfaces.

Extends results to quasi-linear parabolic PDEs like p-Laplace equations.

Abstract

We characterize all the solutions of the heat equation that have their (spatial) equipotential surfaces which do not vary with the time. Such solutions are either isoparametric or split in space-time. The result gives a final answer to a problem raised by M. S. Klamkin, extended by G. Alessandrini, and that was named the Matzoh Ball Soup Problem by L. Zalcman. Similar results can also be drawn for a class of quasi-linear parabolic partial differential equations with coefficients which are homogeneous functions of the gradient variable. This class contains the (isotropic or anisotropic) evolution p-Laplace and normalized p-Laplace equations.