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Planelike minimizers in periodic media
Author(s) -
Caffarelli Luis A.,
de la Llave Rafael
Publication year - 2001
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.10008
Subject(s) - mathematics , bounded function , torus , cover (algebra) , invariant (physics) , integer (computer science) , covering space , plane (geometry) , lamination , mathematical analysis , curvature , pure mathematics , geometry , mathematical physics , layer (electronics) , mechanical engineering , chemistry , organic chemistry , computer science , engineering , programming language
We show that given an elliptic integrand in ℝ d that is periodic under integer translations, and given any plane in ℝ d , there is at least one minimizer of that remains at a bounded distance from this plane. This distance can be bounded uniformly on the planes. We also show that, when folded back to ℝ d /ℤ d , the minimizers we construct give rise to a lamination. One particular case of these results is minimal surfaces for metrics invariant under integer translations. The same results hold for other functionals that involve volume terms (small and average zero). In such a case the minimizers satisfy the prescribed mean curvature equation. A further generalization allows the formulation and proof of similar results in manifolds other than the torus provided that their fundamental group and universal cover satisfy some hypotheses. © 2001 John Wiley & Sons, Inc.