By Luigi Ambrosio

This e-book is dedicated to a thought of gradient flows in areas which aren't inevitably endowed with a usual linear or differentiable constitution. It involves elements, the 1st one pertaining to gradient flows in metric areas and the second dedicated to gradient flows within the area of chance measures on a separable Hilbert area, endowed with the Kantorovich-Rubinstein-Wasserstein distance.

The elements have a few connections, when you consider that the distance of likelihood measures presents an enormous version to which the "metric" conception applies, however the publication is conceived in one of these manner that the 2 elements should be learn independently, the 1st one by means of the reader extra attracted to non-smooth research and research in metric areas, and the second via the reader extra oriented in the direction of the functions in partial differential equations, degree thought and probability.

**Read or Download Gradient flows: In metric spaces and in the space of probability measures PDF**

**Similar differential geometry books**

**Minimal surfaces and Teichmuller theory**

The notes from a collection of lectures writer introduced at nationwide Tsing-Hua college in Hsinchu, Taiwan, within the spring of 1992. This notes is the a part of ebook "Thing Hua Lectures on Geometry and Analisys".

**Complex, contact and symmetric manifolds: In honor of L. Vanhecke**

This publication is concentrated at the interrelations among the curvature and the geometry of Riemannian manifolds. It includes examine and survey articles in response to the most talks brought on the foreign Congress

**Differential Geometry and the Calculus of Variations**

During this publication, we research theoretical and useful elements of computing tools for mathematical modelling of nonlinear structures. a couple of computing thoughts are thought of, corresponding to tools of operator approximation with any given accuracy; operator interpolation ideas together with a non-Lagrange interpolation; tools of process illustration topic to constraints linked to innovations of causality, reminiscence and stationarity; equipment of procedure illustration with an accuracy that's the top inside of a given classification of versions; equipment of covariance matrix estimation;methods for low-rank matrix approximations; hybrid tools in keeping with a mixture of iterative systems and top operator approximation; andmethods for info compression and filtering less than filter out version may still fulfill regulations linked to causality and varieties of reminiscence.

- Differential Harnack inequalities and the Ricci flow
- Dirichlet's principle, conformal mapping, and minimal surfaces
- From Holomorphic Functions to Complex Manifolds
- Supersymmetry and Equivariant de Rham Theory
- Riemann-Finsler Geometry
- Introduction to global analysis, Volume 91

**Additional info for Gradient flows: In metric spaces and in the space of probability measures**

**Sample text**

19). Even in a metric framework, if a curve of maximal slope is a Generalized Minimizing Movement, it exhibits a sort of regularizing eﬀect allowing for a ﬁner description of the diﬀerential equation at each point of the interval, if we consider right derivatives. 2, page. 57]. 15. 5 for some λ ∈ R.

5, |∂φ|(v) is a strong upper gradient. Proof. 19) are simple consequence of the monotonicity of diﬀerence quotients of convex functions. 17). 16), which yields |φ(v) − φ(vn )| ≤ vn − v ξn ∗ + ξ ∗ . 36 Chapter 1. Curves and Gradients in Metric Spaces The inequality w φ(v) − φ(v + w) ≤ ξ, w w ∀w ∈ B \ {0} yields that lφ (v) can be estimated from above by ξ B for any ξ ∈ ∂φ. Assuming that lφ (v) is ﬁnite, to conclude the proof we need only to show the existence of ξ ∈ ∂φ(v) such that ξ B ≤ lφ (v).

Proof. 7) and the diﬀerentiability of φ2 . 17) being φ2 continuous. 4. Curves of maximal slope in Hilbert and Banach spaces 37 that Dφ2 (v) = 0; it follows that φ(w) − φ(v) |∂φ|(v) = lim sup w−v w→v + + φ1 (w) − φ1 (v) |φ2 (w) − φ2 (w)| − lim sup w−v w−v w→v w→v ◦ ◦ = |∂φ1 |(v) = ∂ φ1 (v) ∗ = ∂ φ(v) ∗ . 9), we conclude. Let us rephrase the last conclusion of the previous Corollary, which is quite interesting in the case B does not satisfy the Radon-Nikod´ ym property (as in many examples of rate-independent problems, see [114, 113]).