Anisotropic Isoperimetric Problems and Related Topics

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.

.- Geometric invariants of non-smooth framed curves (by Bevilacqua, Lussardi, and Marzocchi).- Minimal periodic foams with equal cells (by Cesaroni and Novaga).- On a Cheeger--Kohler-Jobin inequality (by Lucardesi, Mazzoleni, and Ruffini).- Isoperimetry on manifolds with Ricci bounded below: overview of recent results and methods (by Pozzetta).- Stochastic homogenization of functionals defined on finite partitions (by Bach and Ruf).- On sets with finite distributional fractional perimeter (by Comi and Stefani).- On a Free-Endpoint Isoperimetric Problem in R^2 (by Alama, Bronsard, and Vriend).- Isoperimetric sets in nonnegative scalar curvature and their role through various concepts of mass (by Benatti and Fogagnolo).- A crystallization result in two dimensions for a soft disc affine potential (by Del Nin and De Luca).

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 Sklodowoska 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.