Our aim is to study the Total Variation Flow (TVF) in metric random walk spaces (MRWS) which include as particular cases: the TVF on locally finite weighted connected graphs. We introduce the concepts of perimeter and mean curvature for subsets of a MRWS. After proving the existence and uniqueness of solutions of the TVF, we study the asymptotic behaviour of those solutions, and for such aim we establish some inequalities of Poincar ́e type. Furthermore, we introduce the concepts of Cheeger and calibrable sets in metric random walk spaces and characterize calibrability by using the 1-Laplacian operator. In connection with the Cheeger cut problem we study the eigenvalue problem whereby we give a method to solve the optimal Cheeger cut problem.

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In this talk I will discuss some recent progress concerning the Navier-Stokes and Euler equations of incompressible fluid. In particular, issues concerning the lack of uniqueness and the effect of physical boundaries on the potential formation of singularity. In addition, I will present a blow-up criterion based on a class of inviscid regularization for these equations.

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With the recent developments of new technologies in biomedical engineering and medicine, the need for new mathematical and numerical methdologies to aid these developments has never been greater. In particular, the design of next generation drug-eluting stents for the treatment of coronary artery disease, and the design of an implantable bioartificial pancreas for the treatment of Type 1 diabetes, hinge on the development of mathematical and computational techniques for solving nonlinear moving boundary problems. In this talk we present recent developments and open problems in the mathematical theory of nonlinear moving boundary problems describing the interaction between viscous, incompressible fluids, and elastic and poroelastic structures. Applications of the mathematical results to coronary angioplasty with stenting, and to bioartificial pancreas design will be shown.

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I will present the recent tools I have developed to prove existence and regularity properties of the critical points of anisotropic functionals. In particular, I will provide the anisotropic extension of Allard's celebrated rectifiability theorem and its applications to the anisotropic Plateau problem. Three corollaries are the solutions to the formulations of the Plateau problem introduced by Reifenberg, by Harrison-Pugh and by Almgren-David. Furthermore, I will present the anisotropic counterpart of Allard's compactness theorem for integral varifolds. To conclude, I will focus on the anisotropic isoperimetric problem: I will provide the anisotropic counterpart of Alexandrov's characterization of volume-constrained critical points among finite perimeter sets. Moreover I will derive stability inequalities associated to this rigidity theorem. 

Some of the presented results are joint works with De Lellis, De Philippis, Ghiraldin, Gioffré, Kolasinski and Santilli.

We will show optimal boundary regularity for bounded positive weak solutions of fast diffusion equations in smooth bounded domains. This solves a problem raised by Berryman and Holland in 1980 for these equations in the subcritical and critical regimes. Our proof of the a priori estimates uses a geometric type structure of the fast diffusion equations, where an important ingredient is an evolution equation for a curvature-like quantity. This is joint work with Jingang Xiong.