00-368 L.A. Caffarelli, R. de la Llave

Plane-like minimizers in periodic media (575K, PDF) Sep 18, 00

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Abstract. We show that given an elliptic integrand $\J$ in $\real^d$ which is periodic under integer translations, given any plane in $\real^d$, there is at least one minimizer of $\J$ which 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 $\real^d/\integer^d$ 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 to formulate and prove similar results in other manifolds than the torus provided that their fundamental group and universal cover satify some hypothesis.

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