- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:06
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
11
2009
4
Critical points via Γ-convergence: general theory and applications
Robert
Jerrard
University of Toronto, TORONTO, ONTARIO, CANADA
Peter
Sternberg
Indiana University, BLOOMINGTON, UNITED STATES
Gamma-convergence, critical points, Allen–Cahn, Ginzburg–Landau
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.
Calculus of variations and optimal control; optimization
General
705
753
10.4171/JEMS/164
http://www.ems-ph.org/doi/10.4171/JEMS/164