The Matzoh Ball Soup Problem: A complete characterization. (January 2016)

• Record Type: Journal Article
• Authors: Magnanini, Rolando; Marini, Michele
• Is Part Of: Nonlinear analysis. Volume 131(2016)
• Page Start: 170
• Page End: 181
• Publication Date: 2016-01
• Subjects: 35K05 -- 35K15 -- 53A10 -- 58J70; Heat equation -- Stationary isothermic surfaces; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.06.022
• Languages: English
• ISSNs: 0362-546X

Abstract

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.