Lecture Notes on Gradient Flows and Optimal Transport

We present a short overview on the strongest variational formulation for gradient flows of geodesically $\lambda$-convex functionals in metric spaces, with applications to diffusion equations in Wasserstein spaces of probability measures. These notes are based on a series of lectures given by the second author for the Summer School "Optimal transportation: Theory and applications" in Grenoble during the week of June 22-26, 2009

Dimensional Reduction and Approximation of Measures and Weakly Differentiable Homeomorphisms

This thesis is devoted to the study of two different problems: the properties of the disintegration of the Lebesgue measure on the faces of a convex function and the existence of smooth approximations of bi-Lipschitz orientation-preserving homeomorphisms in the plane

LECTURE NOTES ON GRADIENT FLOWS AND OPTIMAL TRANSPORT

We present a short overview on the strongest variational formulation for gradi- ent flows of geodesically λ-convex functionals in metric spaces, with applications to diffusion equations in Wasserstein spaces of probability measures. These notes are based on a series of lectures given by the second author for the Summer School “Optimal transportation: Theory and applications” in Grenoble during the week of June 22-26, 2009

A planar bi-Lipschitz extension Theorem

We prove that, given a planar bi-Lipschitz map u defined on the boundary of the unit square, it is possible to extend it to a function v of the whole square, in such a way that v is still bi-Lipschitz. In particular, denoting by L and L˜ the bi-Lipschitz constants of u and v, with our construction one has L˜ ≤ CL4 (C being an explicit geometric constant). The same result was proved in 1980 by Tukia (see [Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), no. 1, 49–72]), using a completely different argument, but without any estimate on the constant L˜. In particular, the function v can be taken either smooth or (countably) piecewise affine