Overview
- is focused on isoperimetric-like problems in non-homogeneous and/or anisotropic media
- offers an updated view of active research topics showcasing new results and detailing open questions
- contains contributions from leading experts or emerging scholars
Part of the book series: Springer INdAM Series (SINDAMS, volume 62)
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About this book
This book contains contributions from speakers at the "Anisotropic Isoperimetric Problems & Related Topics" conference in Rome, held from Sep 5 to 9, 2022.
The classic isoperimetric problem has fascinated mathematicians of all eras, starting from the ancient Greeks, due to its simple statement: what are the sets of a given volume with minimal perimeter? The problem is mathematically well understood, and it plays a crucial role in explaining physical phenomena such as soap bubble shapes.
Variations of the problem, including weighted counterparts with density dependencies, representing inhomogeneity and anisotropy of the medium, broaden its applicability, even in non-Euclidean environments, and they allow for descriptions, e.g., of crystal shapes.
At large, the perimeter's physical interpretation is that of an attractive force; hence, it also appears in describing systems of particles where a balance between attractive and repulsive forces appears. A prominent example is that of Gamow's liquid drop model for atomic nuclei, where protons are subject to the strong nuclear attractive force (represented by the perimeter) and the electromagnetic repulsive force (represented by a nonlocal term). Such a model has been shown to be sound, as it explains the basic characteristics of the nuclei, and it successfully predicts nuclear fission for nuclei with a large atomic number.
Similar energy functionals model various physical and biological systems, showcasing the competition between short-range interfacial and long-range nonlocal terms, leading to pattern formation. The authors mention, e.g., the Ohta–Kawasaki model for microphase separation of diblock copolymers and the Yukawa potential for colloidal systems. Despite diverse systems, the emergence of microphases follows similar patterns, although rigorously proving this phenomenon remains a challenge.
The book collects several contributions within these topics, shedding light on the current state of the art.
Keywords
- shape optimization
- isoperimetric problems with density
- isoperimetric problems in geometric structures
- crystals and periodic structures
- Gamow liquid drop model
- optimal shapes for eigenvalue problems
Editors and Affiliations
About the editors
Valentina Franceschi
Valentina Franceschi is Associate Professor at the University of Padova. Her research interests lie at the interface between analysis and geometry with a particular focus on the study of sub-Riemannian manifolds. She has been Marie Skłodowoska Curie Fellow and Inria Postdoctoral Fellow at LJLL, Sorbonne Université, Lectrice Hadamard at Université Paris-Sud. She is also affiliated to the Padova Neuroscience Center.
Alessandra Pluda
Alessandra Pluda is Tenure-Track Assistant Professor at the University of Pisa. Her research interests are in Geometric Analysis. In particular, she studies geometric evolution equations of second and fourth order with main focus on the description of the long-time behavior of solutions and the analysis of the onset of singularities, employing a mix of PDE and differential geometry techniques.
Giorgio Saracco
Giorgio Saracco is Tenure-Track Assistant Professor at the University of Florence. He previously held postdoctoral positions at the Universities of Erlangen–Nürnberg, Pavia, and at SISSA Trieste. He has been a long-term visiting scholar at the University of Jyväskylä and at the Technical University of München. His interests lie within the realm of calculus of variations and geometric measure theory with a particular focus on sets-depending energies whose leading order term is perimeter-like.
Bibliographic Information
Book Title: Anisotropic Isoperimetric Problems and Related Topics
Editors: Valentina Franceschi, Alessandra Pluda, Giorgio Saracco
Series Title: Springer INdAM Series
Publisher: Springer Singapore
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024
Hardcover ISBN: 978-981-97-6983-4Due: 17 November 2024
Softcover ISBN: 978-981-97-6986-5Due: 17 November 2025
eBook ISBN: 978-981-97-6984-1Due: 17 November 2024
Series ISSN: 2281-518X
Series E-ISSN: 2281-5198
Edition Number: 1
Number of Pages: VII, 159
Number of Illustrations: 9 b/w illustrations, 1 illustrations in colour