A Variational Theory of Convolution-Type Functionals

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 stochastic 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.

Gives an abstract framework for a comprehensive theory of convolution-type functionals Provides an environment and technical tools to frame problems related to multiple-scale variational models Introduces potential applications in different directions from evolution phenomena to data science

Auteur
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 at the 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 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 and Homogenization. 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 stochastic 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 Applied Mathematics, 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.

Contenu

Chapter 1. Introduction.- Chapter 2. Convolution-Type Energies.- Chapter 3. The -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. Stochastic Homogenization.- Chapter 9. Application to Convex Gradient Flows.