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Convergence of Local Minimizers

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

Abstract

The fundamental theorem of Γ-convergence can be generalized when we have strict local minimizers of the Γ-limit, in which case we are often able to deduce the existence and convergence of local minimizers of the converging sequence. This version of the fundamental theorem of Γ-convergence can be coupled with scaling arguments, which may give existence of multiple local minimizers, and in some cases a density result. This approach to showing existence of local minimizers has been introduced by Kohn and Sternberg to show existence of local minimizers in the gradient theory of phase transitions. Their example is included in the notes. Another example shows the existence of local minimizers for oscillating elliptic functionals with a non-convex lower-order perturbation. Finally, we give examples of asymptotically dense sets of local minimizers.

Keywords

  • Strict Local Minimum
  • Gradient Theory
  • Fundamental Theorem
  • Density Results
  • Phase Transition

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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Fig. 5.1
Fig. 5.2
Fig. 5.3
Fig. 5.4

References

  1. Kohn, R.V., Sternberg, P.: Local minimizers and singular perturbations. Proc. Roy. Soc. Edinburgh A 111, 69–84 (1989)

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Appendix

Appendix

The use of Theorem 5.1 for proving the existence of local minimizers, together with Example 5.2 are due to Kohn and Sternberg [1].

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© 2014 Springer International Publishing Switzerland

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Braides, A. (2014). Convergence of Local Minimizers. 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_5

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