Stability theorems

Abstract

We now face the problem of determining conditions under which the minimizing-movement scheme commutes with r-convergence. Let Fe r-converge to F, with initial data xe converging to x0. We have seen in Sect. 8.2 that by suitably choosing e = e(r) the minimizing movement along the sequence Fe from xe converges to a minimizing movement for the limit F from x0. A further issue is whether, by assuming some further properties on Fe, we may deduce that the same thing happens for any choice of e. 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.