Stability Theorems

Abstract

In this chapter we face the problem of determining conditions under which the minimizing-movement scheme commutes with Γ-convergence. Let $F_{\varepsilon }$ Γ-converge to F with initial data $x_{\varepsilon }$ converging to x ₀. In Sect.8.2 it is proved that by suitably choosing $\varepsilon =\varepsilon (\tau )$ the minimizing movement along the sequence $F_{\varepsilon }$ from $x_{\varepsilon }$ converges to a minimizing movement for the limit F from x ₀. A further issue is whether, by assuming some further properties on $F_{\varepsilon }$ we may deduce that the same thing happens for any choice of $\varepsilon$. In order to give an answer we will use results from the theory of gradient flows recently elaborated by Ambrosio, Gigli and Savaré, and by Sandier and Serfaty.

Appendix

The results in Sect. 11.1.1 and part (ii) of Theorem 11.1 are a simplified version of the analogous results for geodesic-convex energies in metric spaces, that can be found in the notes by Ambrosio and Gigli [1]. Example 11.2 is a simplified version of a result by Braides et al. [2].

The result by Sandier and Serfaty (with weaker hypotheses than those reported here) is contained in the seminal paper [4]. An account of their approach is contained in the notes by Serfaty [6].

The convergence of stable points has been considered by Serfaty in [5] and further analyzed by Jerrard and Sternberg in [3].