2024/06/01 by Indrei, Emanuel, Karakhanyan, Aram
#28A75 #49Jxx #49Q20 #49Sxx #52Axx #80Axx #82B30 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2406.00241
In this paper we completely settle the Almgren problem in \mathbb R3 under some generic conditions on the potential and tension functions. The problem, among other things, appears in classical thermodynamics when one is to understand if minimizing the free energy with convex potential and under a mass constraint generates a convex crystal. Our new idea in proving a three-dimensional convexity theorem is to utilize a stability theorem when m is small, convexity when m is small, and the first variation PDE with a new maximum principle approach.