Isoperimetric Clusters on Riemannian and Finsler Manifolds: Properties and Applications via the Photographic Method

• Grant number: 25/07027-9
• Support Opportunities: Scholarships abroad - Research Internship - Post-doctor
• Start date: July 14, 2025
• End date: July 13, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Stefano Nardulli
• Grantee: Raoní Cabral Ponciano
• Supervisor: Aldo Pratelli
• Host Institution: Centro de Matemática, Computação e Cognição (CMCC). Universidade Federal do ABC (UFABC). Ministério da Educação (Brasil). Santo André , SP, Brazil
• Institution abroad: Università di Pisa, Italy
• Associated to the scholarship: 23/07697-9 - Multiplicity solutions and qualitative properties for some classes of nonlinear elliptic equations and Sobolev embeddings on limit case, BP.PD

Abstract

This BEPE project outlines the scientific articles we plan to develop under the guidance of the esteemed researcher Aldo Pratelli from the Università di Pisa. Our goal is to bring to S\~ao Paulo the advanced techniques from his expertise in Geometric Measure Theory. Specifically, we aim to establish diameter bounds for small-volume isoperimetric clusters on Riemannian and Finsler manifolds. Furthermore, we intend to extend these results to the setting of weighted perimeters, thereby obtaining multiple solutions for the Allen-Cahn system under the assumptions of a small temperature parameter and small volume. Additionally, under the supervision of the renowned professor Vieri Benci, we seek to develop results on ultrafunctions, demonstrating the existence of ultrafunction solutions to problems that are ill-posed in the distributional framework.