By Luigi Ambrosio

This publication is dedicated to a concept of gradient flows in areas which aren't inevitably endowed with a common linear or differentiable constitution. It contains components, the 1st one touching on gradient flows in metric areas and the second dedicated to gradient flows within the house of chance measures on a separable Hilbert area, endowed with the Kantorovich-Rubinstein-Wasserstein distance.

The components have a few connections, in view that the gap of chance measures presents a huge version to which the "metric" idea applies, however the booklet is conceived in one of these approach that the 2 components could 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 by way of the reader extra oriented in the direction of the purposes in partial differential equations, degree conception and probability.

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**Example text**

1 hold and let Λ be a family of partitions with inf τ ∈Λ |τ | = 0. 8) then there exist a sequence (τ k ) ⊂ Λ with |τ k | ↓ 0 and a limit curve u ∈ 2 ACloc ([0, +∞); S ) such that Uτ k (t) σ ∀ t ∈ [0, +∞). 9) σ In particular, if Uτ0k u0 and φ(Uτ0k ) → φ(u0 ) as k → ∞, then u(0+) = u0 and u ∈ GM M (Φ; u0 ), which is therefore a nonempty set. We prove this proposition in the next Section 3. 4 (p-estimates). 3 still hold (with diﬀerent constants) simply replacing 2 with p: thus the p limiting curve belongs to ACloc ([0, +∞); S ).

10b) with respect to the weak Lp -topology σ. e. in Rd so that η = F (v) and therefore ξ ∈ ∂φ(v). 4 The (geodesically) convex case In this section we will consider a notion of convexity along classes of curves in the metric space S : a particular attention is devoted to functionals φ which are convex along the geodesics of the metric space S . Let us ﬁrst introduce the relevant deﬁnitions. 50 Chapter 2. 1 (λ-convexity along curves). A functional φ : S → (−∞, +∞] is called convex on a curve γ : t ∈ [0, 1] → γt ∈ S if φ(γt ) ≤ (1 − t)φ(γ0 ) + tφ(γ1 ) ∀t ∈ [0, 1].

4). ∂ ◦ φ(v) is the subset of elements of minimal (dual) norm in ∂φ(v), which reduces to a single point if the dual norm of B is strictly convex. Notice that |∂φ|(v) = lim sup w→0 φ(v) − φ(v + w) w ≤ lim sup ξ, w w w→0 ≤ ξ ∗ ∀ ξ ∈ ∂φ(v). 34 Chapter 1. 5 that the map v → ∂ ◦ φ(v) ∗ is a weak upper gradient for φ. e. 10) as the solution of a suitable doubly nonlinear diﬀerential inclusion: in the case when S is a reﬂexive Banach space and φ is convex, these kind of evolution equations have been studied in [53, 52]; we refer to these contributions and to [148] for many examples of partial diﬀerential equations which can be studied by this abstract approach.