Calculus of Variations and Geometric Measure Theory

D. Zucco

Ahlfors regularity of continua that minimize maxitive set functions

created by zucco on 15 Mar 2024



Inserted: 15 mar 2024
Last Updated: 15 mar 2024

Year: 2024

ArXiv: 2403.09251 PDF


The primary objective of this paper is to establish the Ahlfors regularity of minimizers of set functions that satisfy a suitable maxitive condition on disjoint unions of sets. Our analysis focuses on minimizers within continua of the plane with finite one-dimensional Hausdorff measure. Through quantitative estimates, we prove that the length of a minimizer inside the ball centered at one of its points is comparable to the radius of the ball. By operating within an abstract framework, we are able to encompass a diverse range of entities, including spectral functionals defined in terms of the eigenvalues of elliptic operators, the inradius, and the maximum of the torsion function. These entities are of interest for several applications, such as structural engineering, urban planning, and quantum mechanics.