Convergence of Local Minimizers

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.

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].