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Stability Theorems

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2094)

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.

Keywords

  • Stability Theorem
  • Gradient Flow
  • Initial Data
  • Convex Energy
  • Continuous-time Interpolation

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Ambrosio, L., Gigli, N.: A user’s guide to optimal transport. In: Piccoli, B., Rascle, M. (eds.) Modelling and Optimisation of Flows on Networks. Lecture Notes in Mathematics, pp. 1–155. Springer, Berlin (2013)

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  4. Sandier, E., Serfaty, S.: Gamma-convergence of gradient flows and application to Ginzburg–Landau. Comm. Pure Appl. Math. 57, 1627–1672 (2004)

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  5. Serfaty, S.: Stability in 2D Ginzburg–Landau passes to the limit. Indiana Univ. Math. J. 54, 199–222 (2005)

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Appendix

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].

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Braides, A. (2014). Stability Theorems. In: Local Minimization, Variational Evolution and Γ-Convergence. Lecture Notes in Mathematics, vol 2094. Springer, Cham. https://doi.org/10.1007/978-3-319-01982-6_11

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