By Eduard Feireisl
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7, we will assume with no loss of generality that |μ | ∈ L∞ (I ) in the proof of the first statement. To fix the ideas, we also assume that I = (0, 1). First of all we show that for every ϕ ∈ Cc∞ (Rd ) the function t → μt (ϕ) is absolutely continuous, and its derivative can be estimated with the metric derivative of μt . Indeed, for s, t ∈ I and μst ∈ Γo (μs , μt ) we have, using the Hölder inequality, μt (ϕ) − μs (ϕ) = Rd ϕ(y) − ϕ(x) dμst Lip(ϕ)W2 (μs , μt ), 32 L. Ambrosio and G. Savaré whence the absolute continuity follows.
Therefore W22 μ2→3 , μ1 = min 5t 2 − 7t + t 13 2 9 , 5t − 3t + 2 2 has a concave cusp at t = 1/2 and therefore is not λ-convex along the geodesic μ2→3 for t any λ ∈ R. 2. Examples of convex functionals in P2 (Rd ) In this section we introduce the main classes of geodesically convex functionals. 4 (Potential energy). , V (x) −A − B|x|2 ∀x ∈ Rd for some A, B ∈ R+ . 6) In P2 (Rd ) we define V(μ) := Rd V (x) dμ(x). 2 gives that V is lower semicontinuous in P2 (Rd ). t. narrow convergence. The following simple proposition shows that V is convex along all interpolating curves induced by admissible plans; choosing optimal plans one obtains in particular that V is convex along geodesics.
56) μ where tμt+h is the unique optimal transport map between μt and μt+h , and t lim h→0 W2 (μt+h , (i + hvt )# μt ) = 0. 57) P ROOF. Let D ⊂ Cc∞ (Rd ) be a countable set with the following property: for any integer R > 0 and any ϕ ∈ Cc∞ (Rd ) with supp ϕ ⊂ BR there exist (ϕn ) ⊂ D with supp ϕn ⊂ BR and ϕn → ϕ in C 1 (Rd ). We fix t ∈ I such that W2 (μt+h , μt )/|h| → |μ |(t) = vt L2 (μt ) and lim h→0 μt+h (ϕ) − μt (ϕ) = h Rd ∇ϕ, vt dμt ∀ϕ ∈ D. e. t ∈ I . Set μ sh := −i tμt+h t h and fix ϕ ∈ D and a weak limit point s0 of sh as h → 0.
Handbook of Differential Equations - Evolutionary Equations by Eduard Feireisl