A Variational Theory of Convolution-Type Functionals

by Roberto Alicandro, Nadia Ansini, Andrea Braides, Andrey Piatnitski, Antonio Tribuzio

Overview

This book provides a general treatment of a class of functionals modelled on convolution energies with kernel having finite p-moments. A general asymptotic analysis of such non-local functionals is performed, via Gamma-convergence, in order to show that the limit may be a local functional representable as an integral. Energies of this form are encountered in many different contexts and the interest in building up a general theory is also motivated by the multiple interests in applications (e.g. peridynamics theory, population dynamics phenomena and data science). The results obtained are applied to periodic and shastic homogenization, perforated domains, gradient flows, and point-clouds models.

This book is mainly intended for mathematical analysts and applied mathematicians who are also interested in exploring further applications of the theory to pass from a non-local to a local description, both in static problems and in dynamic problems.

Product Details

ISBN-13:	9789819906840
Publisher:	Springer Nature Singapore
Publication date:	05/02/2023
Series:	SpringerBriefs on PDEs and Data Science
Edition description:	1st ed. 2023
Pages:	116
Product dimensions:	6.10(w) x 9.25(h) x (d)

About the Author

Roberto Alicandro is professor of Mathematical Analysis at Università di Cassino e del Lazio meridionale. He is an expert in the Calculus of Variations and Homogenization and his results have applications in different fields, including atomistic-to-continuum limits for nonlinear models in material science, phase transition problems, topological singularities and defects in materials. He is the author of a monograph on Discrete Variational Problems with Andrea Braides and other co-authors.

Nadia Ansini is professor of Mathematical Analysis atthe Department of Mathematics, Sapienza University of Rome. She is an expert in the Calculus of Variations, Homogenization and Multiple-scale models in mathematical materials science with subjects ranging from perforated domains, thin films, phase transitions, and variational evolution problems. She was awarded with two Marie Sklodowska-Curie Fellowships in 2000 and 2012. She is Lise Meitner visiting professor at Lund University (Sweden, 2022-2025).

Andrea Braides is professor of Mathematical Analysis at SISSA, Trieste, on leave from the University of Rome Tor Vergata. He is an expert in the Calculus of Variations andHomogenization. He is the author of several monographs in the fields of Gamma-convergence and Discrete Variational Problems. He was an invited speaker at the 2014 International Congress of Mathematicians in Seoul in the section Mathematics in Science and Technology.

Andrey Piatnitsky is an expert in the Calculus of Variations and in Partial Differential Equations, specializing in the homogenization of both deterministic and shastic energies and operators, and singularly perturbed operators. He has been the invited speaker to major international conferences on these subjects. He and his co-authors produced a monograph on Homogenization.

Antonio Tribuzio is a research fellow at the Institute for AppliedMathematics, Heidelberg University. His field of expertise is the Calculus of Variations. He worked, among others, on the relation between De Giorgi's Minimizing Movements and Gamma-convergence, discrete evolutions and scaling behaviour of energies related to Shape-Memory Alloys.

Table of Contents

Chapter 1. Introduction.- Chapter 2. Convolution-Type Energies.- Chapter 3. The Γ-limit of a Class of Reference Energies.- Chapter 4. Asymptotic Embedding and Compactness Results.- Chapter 5. A Compactness and Integral-Representation Result.- Chapter 6. Periodic Homogenization.- Chapter 7. A Generalization and Applications to Point Clouds.- Chapter 8. Shastic Homogenization.- Chapter 9. Application to Convex Gradient Flows.