Boundary regularity for area minimizing currents

• Grant number: 18/22938-4
• Support Opportunities: Scholarships abroad - Research
• Start date: January 15, 2020
• End date: January 14, 2021
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Stefano Nardulli
• Grantee: Stefano Nardulli
• Host Investigator: Camillo de Lellis
• Host Institution: Centro de Matemática, Computação e Cognição (CMCC). Universidade Federal do ABC (UFABC). Ministério da Educação (Brasil). Santo André , SP, Brazil
• Institution abroad: Princeton University, United States

Abstract

The Plateau's problem investigates those surfaces of least area spanning a given contour. The very formulation of the Plateau's problem has proved to be a quite challenging mathematical question. In particular, how general are the surfaces that one should consider? What is the correct concept of ``spanning'' and the correct concept of "$m$-dimensional volume'' that one should use? We believe that there are no final answers to these two questions: many different significant ones have been given in the history of the subject and, depending upon the context, the features of one formulation might be considered more important than those of the others. In this project we focus on the most successful ''functional-analytic formulation'': the surfaces are viewed as objects acting on a given (linear) space of smooth test functions, via integration, specifically we will deal with Federer and Fleming's theory of integral currents, which had a precursor in codimension one in the De Giorgi's pioneering theory of sets of finite perimeter. The aim of this project is to explore the boundary regularity theory in codimension higher than $1$, which is a widely open problem. (AU)

Scientific publications

(4)

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

MUNOZ FLORES, ABRAHAM ENRIQUE; NARDULLI, STEFANO. Generalized Compactness for Finite Perimeter Sets and Applications to the Isoperimetric Problem. JOURNAL OF DYNAMICAL AND CONTROL SYSTEMS, v. 28, n. 1, p. 59-69, JAN 2022. (18/22938-4)

BENCI, VIERI; NARDULLI, STEFANO; PICCIONE, PAOLO; ACEVEDO, LUIS EDUARDO OSORIO. Lusternik-Schnirelman and Morse Theory for the Van der Waals-Cahn-Hilliard equation with volume constraint. NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, v. 220, p. 29-pg., 2022-03-18. (16/23746-6, 21/05256-0, 17/13155-3, 18/22938-4)

DE LELLIS, CAMILLO; NARDULLI, STEFANO; STEINBRUECHEL, SIMONE. Uniqueness of boundary tangent cones for 2-dimensional area-minimizing currents. NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, v. 230, p. 10-pg., 2023-02-15. (18/22938-4, 21/05256-0)

NARDULLI, STEFANO; RUSSO, FRANCESCO G.. On the Hamilton's isoperimetric ratio in complete Riemannian manifolds of finite volume. JOURNAL OF FUNCTIONAL ANALYSIS, v. 280, n. 4, FEB 15 2021. (18/22938-4)