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Description
This volume is based on the talks presented at the "New Frontiers in Homogenization and Fractional Calculus" conference, held at CRM-Barcelona on March 24 25, 2025. This event was organized to celebrate the 50th anniversary of the mathematical technique of Gamma-convergence, introduced by Ennio De Giorgi and Tullio Franzoni in 1975. Originally developed for applied purposes, this technique remains a fundamental tool in the study of various scientific phenomena, such as homogenization, phase transitions, and the asymptotic analysis of partial differential equations. At the same time, there has been a growing interest in the study of nonlocal problems, particularly due to their relevance in probability theory. In addition to representing a particularly challenging area of mathematics, these problems have become increasingly relevant in material science applications.
The contributions in this volume cover a wide range of topics and reflect the main areas of current research in variational convergence and nonlocal analysis. Additionally, as these techniques extend beyond pure analytical theory, the contributions explore numerical applications. This volume is aimed at students and researchers in mathematics, physics, and engineering who are interested in exploring recent developments in the theories of homogenization and fractional calculus.
Chapter 1. Singular-perturbation problems in fractional Sobolev spaces recent results.- Chapter 2. A Survey on the Div-Curl Lemma and Some Extensions to Fractional Sobolev Spaces.- Chapter 3. The Polya-Szego principle in the fractional setting: a glimpse on nonlocal functional inequalities.- Chapter 4. Numerical Approximation of the logarithmic Laplacian via sinc-basis.- Chapter 5. A survey on the resolvent convergence.- Chapter 6. Fractional Sobolev spaces via interpolation, and applications to mixed local-nonlocal operators.- Chapter 7. Some results about strongly degenerate parabolic equations and forward-backward parabolic equations.- Chapter 8. A survey on anisotropic integral representation results.
Alberto Maione is a María de Maeztu Postdoctoral Fellow at the Centre de Recerca Matemàtica (CRM) in Barcelona, where he works in the Analysis & PDEs research group. He obtained his PhD in 2020 from the University of Trento and the University of Verona, with a thesis on Calculus of Variations and Partial Differential Equations entitled Variational Convergences for Functionals and Differential Operators Depending on Vector Fields . In 2023, the thesis was recognized as the third best in applied analysis by the University of Padua. His research focuses on variational convergences, a powerful tool in analysis for approximating models in physics, engineering, and material science. Since 2022, he has taught these methods at leading universities in Germany and Spain. He has co-authored several peer-reviewed publications, including an in-depth study of sub-Riemannian geometry and fractional operators. In addition to his research, Dr. Maione has been invited to present his work at several international conferences. He has also been actively involved in organizing international scientific events in Italy, Germany and Spain and is currently leading an international research-in-pairs project, focusing on homogenization and fractional calculus.
Joaquim Duran i Lamiel is a María de Maeztu Doctoral Fellow at the Centre de Recerca Matemàtica (CRM) in Barcelona, where he works in the PDEs research group. In 2023, he graduated in physics and mathematics from the University of Barcelona, obtaining the August Palanques Award for the best academic record in the Mathematics degree. His research focuses on the spectral analysis of relativistic models arising in quantum mechanics.
