Geometric variational problems: existence, regularity and geometrical characterization of the solutions

• Grant number: 21/05256-0
• Support Opportunities: Research Grants - Young Investigators Grants
• Start date: November 01, 2021
• End date: October 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Stefano Nardulli
• Grantee: Stefano Nardulli
• Host Institution: Centro de Matemática, Computação e Cognição (CMCC). Universidade Federal do ABC (UFABC). Ministério da Educação (Brasil). Santo André , SP, Brazil
• Associated researchers: Dario Corona ; Fabio Giannoni ; João Henrique Santos de Andrade ; Márcio Fabiano da Silva ; Paolo Piccione ; Raoní Cabral Ponciano ; Reinaldo Resende de Oliveira
• Associated research grant(s): 23/08246-0 - Breaking new ground in the ACH equation: a research partnership between São Paulo and UNICAM, AP.R SPRINT
• Associated scholarship(s): 24/16321-5 - Logical and categorical methods in calculus of variations: an Introduction., BP.IC; 24/19166-0 - Deep Currents and the Plateau problem, BP.IC; 24/14876-0 - Currents and Q valued functions., BP.MS; 24/14437-6 - Study and Generalization of the Weierstrass Approximation Theorem and Stone's Generalization., BP.IC

Abstract

Geometric variational problems are among the most fascinating problems of the calculus of variations. A lot among the most interesting problems of differential geometry are variational in nature. The solution to such problems describes equilibrium shapes of physical systems or give selected representative belonging to particular class of homology and homotopy. To study existence multiplicity and geometric characterization of such solutions is of fundamental in pure and applied mathematics. This project have as a primary goal do improve our understanding about them developing new methods and approaches. Among the problems treated in this project we have the isoperimetric problem, the oriented Plateau's problem, fase transition problems with emphasis in the Cahn-Hilliard equation. All these problems will be treated in smooth and non-smooth metric measures spaces using the most recent technics of non-smooth analysis in metric measure spaces combined with the sophisticated theory of regularity for $Q$-valued functions of Almgren. great relevance will be given to the theory of existence and multiplicity of geometric variational problems with lack of compactness in non compact ambient spaces with bounded geometry. In these spaces we will give a quite satisfactory general answer. (AU)

Scientific publications

(7)

(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)

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)

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)

NARDULLI, STEFANO; RESENDE, REINALDO. Density of the boundary regular set of 2d area minimizing currents with arbitrary codimension and multiplicity. ADVANCES IN MATHEMATICS, v. 455, p. 78-pg., 2024-08-21. (21/05256-0)

BENCI, VIERI; CORONA, DARIO; NARDULLI, STEFANO; ACEVEDO, LUIS EDUARDO OSORIO; PICCIONE, PAOLO. Lusternik-Schnirelman and Morse Theory for the Van der Waals-Cahn-Hilliard equation with volume constraint (vol 220, 112851, 2022). NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, v. 238, p. 9-pg., 2023-09-23. (22/13010-3, 21/05256-0, 16/23746-6)

DE ROSA, ANTONIO; RESENDE, REINALDO. Boundary regularity for anisotropic minimal Lipschitz graphs. Communications in Partial Differential Equations, v. 49, n. 1-2, p. 23-pg., 2023-12-20. (21/05256-0)

DE LELLIS, CAMILLO; NARDULLI, STEFANO; STEINBRUCHEL, SIMONE. AN ALLARD-TYPE BOUNDARY REGULARITY THEOREM FOR 2d MINIMIZING CURRENTS AT SMOOTH CURVES WITHARBITRARY MULTIPLICITY. PUBLICATIONS MATHEMATIQUES DE L IHES, v. 140, n. 1, p. 118-pg., 2024-02-21. (18/22938-4, 21/05256-0)

ANTONELLI, GIOACCHINO; NARDULLI, STEFANO; POZZETTA, MARCO. The isoperimetric problem via direct method in noncompact metric measure spaces with lower Ricci bounds. ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS, v. 28, p. 32-pg., 2022-08-12. (21/05256-0)