Journal of the European Mathematical Society


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Volume 11, Issue 4, 2009, pp. 705–753
DOI: 10.4171/JEMS/164

Critical points via Γ-convergence: general theory and applications

Robert L. Jerrard[1] and Peter Sternberg[2]

(1) Department of Mathematics, University of Toronto, 40 St. George Street, M5S 2E4, TORONTO, ONTARIO, CANADA
(2) Department of Mathematics, Indiana University, IN 47405, BLOOMINGTON, UNITED STATES

It is well-known that Γ-convergence of functionals provides a tool for studying global and local minimizers. Here we present a general result establishing the existence of critical points of a Γ-converging sequence of functionals provided the associated Γ-limit possesses a nondegenerate critical point, subject to certain mild additional hypotheses. We then go on to prove a theorem that describes suitable nondegenerate critical points for functionals, involving the arclength of a limiting singular set, that arise as Γ-limits in a number of problems. Finally, we apply the general theory to prove some new results, and give new proofs of some known results, establishing the existence of critical points of the 2d Modica–Mortola (Allen–Cahn) energy and 3d Ginzburg–Landau energy with and without magnetic field, and various generalizations, all in a unified framework.

Keywords: Gamma-convergence, critical points, Allen–Cahn, Ginzburg–Landau

Jerrard R, Sternberg P. Critical points via Γ-convergence: general theory and applications. J. Eur. Math. Soc. 11 (2009), 705-753. doi: 10.4171/JEMS/164