Mathematical Analysis Seminars @ DISIM

Title: A variational approach to topological singularities through Mumford-Shah type functionals

Abstract: We will present variational approaches to the analysis of topological singularities in the plane, starting from the - nowadays - classical Ginzburg-Landau (GL) model and core- radius (CR) approach. We will introduce a third approach inspired by the Mumford-Shah functional used in the context of image segmentation. Within our framework, the order parameter is an SBV map taking values in the unit sphere of the plane; the bulk energy is the squared L2 norm of the approximate gradient whereas the penalization term is given by the length of the jump set, scaled by a small parameter. After providing a notion of Jacobian determinant for SBV maps, we show that at any logarithmic scale our functional is "variationally equivalent" to the "standard" (CR) and (GL) models. Joint work with Vito Crismale, Nicolas Van Goethem and Riccardo Scala.

Title: The black hole stability problem in general relativity

Abstract: Black holes are perhaps the most spectacular theoretical prediction of Einstein's theory of general relativity. A mathematical proof of their stability as solutions to the Einstein equations remains a fundamental open problem in the subject. The talk will outline the formulation, major difficulties, and implications of the problem, including its connections with recent advances in the experimental observation of these objects.

Title: TBA

Abstract: TBA

Title: Quantitative vanishing viscosity approximation of fully nonlinear, non-convex, Hamilton-Jacobi equations with Hölder data.

Abstract: We discuss new quantitative estimates of the vanishing viscosity process for evolutionary Hamilton-Jacobi PDEs that are neither concave nor convex in the gradient and Hessian entries. I will describe a novel approach that exploits the regularizing properties of sup/inf-convolutions for viscosity solutions combined with the comparison principle. This method provides explicit sharp constants without assuming differentiability properties neither on solutions nor on the Hamiltonian. This is a joint work with Alekos Cecchin (Padova).