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Functions of Bounded Variation and Free Discontinuity Problems
Measure Theory Basic Geometric Measure Theory Functions of bounded variation Special functions of bounded variation Semicontinuity in BV The Mumford-Shah functional Minimisers of free continuityExpand
Gradient Flows: In Metric Spaces and in the Space of Probability Measures
Notation.- Notation.- Gradient Flow in Metric Spaces.- Curves and Gradients in Metric Spaces.- Existence of Curves of Maximal Slope and their Variational Approximation.- Proofs of the ConvergenceExpand
Currents in metric spaces
C o n t e n t s Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ] 1. Notation and preliminary results . . . . . . . . . . . . . . . . . . . . 7 2. Metric functionals .Expand
Approximation of functional depending on jumps by elliptic functional via t-convergence
We show how it is possible to approximate the Mumford-Shah (see [29]) image segmentation functional by elliptic functionals defined on Sobolev spaces. The heuristic idea is to consider functionalsExpand
Lecture Notes on Optimal Transport Problems
1 Some elementary examples 2 Optimal transport plans: existence and regularity 3 The one dimensional case 4 The ODE version of the optimal transport problem 5 The PDE version ofExpand
Existence theory for a new class of variational problems
Existence theory for a new class of variational problems.
A User’s Guide to Optimal Transport
This text is an expanded version of the lectures given by the first author in the 2009 CIME summer school of Cetraro. It provides a quick and reasonably account of the classical theory of optimalExpand
Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below
This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces $(X,\mathsf {d},\mathfrak {m})$. Our main results are: A generalExpand
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