On the Three-Dimensional Shape of a Crystal

Emanuel Indrei () and Aram Karakhanyan
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Emanuel Indrei: Department of Mathematics, Kennesaw State University, Marietta, GA 30060, USA
Aram Karakhanyan: School of Mathematics, The University of Edinburgh, Peter Tait Guthrie Road, Edinburgh EH9 3FD, UK

Mathematics, 2025, vol. 13, issue 4, 1-13

Abstract: The purpose of this paper is to investigate the Almgren problem in R 3 under generic conditions on the potential and tension functions which define the free energy. This problem appears in classical thermodynamics when one seeks to understand whether minimizing the free energy with convex potential in the class of sets of finite perimeter under a mass constraint generates a convex minimizer representing a crystal. Our new idea in proving a three-dimensional convexity theorem is to utilize convexity and a stability theorem when the mass is small, as well as a first-variation partial differential equation along with a new maximum principle approach.

Keywords: free energy; crystals; thermodynamics; surface energy; potential energy; convex; first variation; Almgren’s problem; isoperimetric (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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