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Planelike minimizers in periodic media. (English) Zbl 1036.49040
Summary: We show that given an elliptic integrand \({\mathcal J}\) in \(\mathbb{R}^d\) that is periodic under integer translations, and given any plane in \(\mathbb{R}^d\), there is at least one minimizer of \({\mathcal J}\) 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 \(\mathbb{R}^d/\mathbb{Z}^d\), the minimizers we construct give rise to a lamination. One particular case of these results is the 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.

MSC:
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
49J10 Existence theories for free problems in two or more independent variables
49N60 Regularity of solutions in optimal control
58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
49Q05 Minimal surfaces and optimization
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