Classical minimal surfaces arise from the minimisation of a perimeter functional: for instance, given a domain in the Euclidean space, one minimises the boundary measure in the domain of sets with prescribed conditions along the boundary of the domain. Besides its clear geometric interest, this topic is related to many important problems in partial differential equations and applied maths, such as phase coexistence models and capillarity. While the classical model is purely "local" (the perimeter of a set in a domain only depends on the set in the domain itself), recently, a lot of interest has arisen for "nonlocal" models, in which far-away contributions may significantly influence the shape of the objects taken into account.
In this spirit, we discuss some recent results on a geometric problem of nonlocal type, consisting of the minimisation of a fractional version of the perimeter. In particular, we will discuss the rather unexpected behaviour of the nonlocal minimisers at the boundary of the reference domain.
About the speaker
Serena Dipierro received her PhD in Mathematical Analysis at the International School for Advanced Studies (SISSA, Trieste) in 2012. After PostDoc positions at the Universidad de Chile and University of Edinburgh, and a Humboldt Fellowship, she held permanent positions at the University of Melbourne and the Università di Milano. Since August 2018, she is an Associate Professor at the University of Western Australia.
Serena's research focuses on partial differential equations, free boundary problems, nonlocal equations, calculus of variations, nonlinear analysis, and applications.