Responsabile scientifico

• BUOSO Davide

Partecipanti al progetto

• FERRERO Alberto

Titolo del progetto

• Perturbation problems and asymptotics for elliptic differential equations: variational and potential theoretic methods

Ente finanziatore

• MUR - Ministero dell'Università e Ricerca

Programma di finanziamento

• PRIN - Progetti di Ricerca di Rilevante Interesse Nazionale (dal 2020)

Bando di riferimento

• PRIN 2022

Anno di presentazione del progetto

• 2022

Anno di approvazione del progetto

• 2023

Project ID / numero contratto

• 2022SENJZ3_001

Durata del progetto

• 24 mesi

Inizio del progetto

• 28/09/2023

Fine del progetto

• 27/09/2025

Ente capofila

• UNIUPO - Universita Degli Studi del Università del Piemonte Orientale "Amedeo Avogadro"

Altri partner del progetto

• UNIBAS - Università degli studi della Basilicata
• UNIPD - Università degli Studi di Padova
• UNIROMA1 - Università di Roma La Sapienza
• UNIVE - Università degli studi di Venezia Ca' Foscari

Progetto approvato - Contributo unità di ricerca del Dipartimento

• €57.608,00

Sustainable Development Goals - Agenda 2030

Keyword

• Spectral geometry
• Perturbation of PDEs
• singular perturbations
• integral representation of solutions
• shape sensitivity analisys
• series solutions to PDEs

Stato del progetto

• Approvato

Abstract

• In this project we will consider perturbation and asymptotic problems for elliptic differential equations. We will study old and new problems, many of them of actual interest, and apply innovative methodologies arising from new points of view both within analysis and geometry. We will consider several different types of perturbation: domain (regular and singular perturbations for electromagnetic, degenerate, Steklov, nonlinear and higher order problems, corner singularities, etc.), mass and geometry (eigenvalue bounds and optimization), coefficients (regularity and stability, constant/nonconstant cases). For each particular situation we will use suitably designed approaches, either improving on well-established ones, or coming from new perspectives that we will highlight, develop, and fine-tune. In particular, we plan to exploit the interplay between potential theory and calculus of variations and, on a higher scale, we will involve more prominently geometric ideas in unprecedented ways: we will not only study perturbation and asymptotic problems in Riemannian settings, but also apply geometric techniques for the study of problems in Euclidean spaces. Apart from actual perturbation problems, we will also consider more abstract, foundational questions that will be necessary to improve the understanding of the geometrical and functional structure, such as: the role of the mass from a geometric point of view; domain perturbation in a general Riemannian setting; reducible operators for solving general BVPs; numerical computation of potentials; regularity properties of layer potentials. The research team consists of 5 Research Units (RU): Eastern Piedmont, Padova, Ca' Foscari, Sapienza, and Basilicata. The partners combine experts from different areas of mathematical analysis such as Spectral Theory, Potential Theory, Functional Analysis, Integral Operators, and Geometric Analysis. This combination provides the specific expertises necessary for a successful implementation of the project. Existing connections among the RU will be reinforced and new ones will be created, both within the project and outside. We will disseminate the results within our communities and try to reach for new ones, to amplify the spread of the project outcomes. We will organize an international conference on the themes of perturbations, asymptotics, and related tools. We will open post-doctoral fellowships to work on the subject and deliver PhD classes to engage young students. We will design activities for public engagement of schools and non-specialist audiences, with the aim to create awareness of the importance of science for society. Although this project is more theoretical-oriented, we remark that there are a number of relevant real-world applications directly linked with the various problems we intend to work on (e.g., bridges design, electromagnetic problems, machine learning, 3D shape analysis, medical image analysis, computer graphics, composite materials, etc.).