Work

The (Self-Similar, Variational) Rolling Stones

Year: 2024

Type: article

Abstract: The interplay between variational functionals and the Brunn–Minkowski Theory is a well-established phenomenon widely investigated in the last thirty years. In this work, we prove the existence of solutions to the even logarithmic Minkowski problems arising from variational functionals, such as the first eigenvalue of the Laplacian and the torsional rigidity. In particular, we lay down a blueprint showing that the same result holds for more generic functionals by adapting the volume case from Böröczky, Lutwak, Yang, and Zhang. We show how these results imply the existence of self-similar solutions to variational flow problems à la Firey’s worn stone problem.

Source: International Mathematics Research Notices

DOI: 10.1093/imrn/rnae019

Authors: Dylan Langharst, Jacopo Ulivelli

Institutions: Institut de Mathématiques de Jussieu-Paris Rive Gauche, Sorbonne Université, Sapienza University of Rome

Language: English

Topic: Point processes and geometric inequalities

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, Fondation Sciences Mathématiques de Paris

Awards: Institute for Computational and Experimental Research in Mathematics, Harmonic Analysis in Convex Geometry