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The existence of gaps in minimal foliations. (English) Zbl 0645.58017
Harmonic functions \(v: {\mathbb{R}}^ n\to {\mathbb{R}}\) minimize the Dirichlet integral \(\int | v_ x|\) 2 dx with respect to compactly supported variations of v. So, for fixed slope \(\alpha\in {\mathbb{R}}^ n\) the graphs of the affine functions \(v(x)=\alpha \cdot x+v_ 0\), \(v_ 0\in {\mathbb{R}}\), constitute a minimal foliation with respect to the Dirichlet integral. Following J. Moser [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3, 229-272 (1986; Zbl 0609.49029)] we generalize the Dirichlet integrands \(| v_ x|^ 2\) to integrands \(F: {\mathbb{R}}^ n\times {\mathbb{R}}\times {\mathbb{R}}^ n\to {\mathbb{R}}\) and consider F-minimal functions \(v: {\mathbb{R}}^ n\to {\mathbb{R}}\) defined by the property that \[ \int_{{\mathbb{R}}^ n}\{F(x,v+\phi,v_ x+\phi_ x)- F(x,v,v_ x)\}dx\geq 0 \] for all compactly supported \(\phi: {\mathbb{R}}^ n\to {\mathbb{R}}\). Our main hypothesis is that F be \({\mathbb{Z}}\)-periodic in the first \(n+1\) variables. Under appropriate growth conditions on F, Moser proves the existence of sets \(\mu_{\alpha}^{rec}(F)\) of F-minimal functions which are natural generalizations of the sets \(\mu_{\alpha}^{rec}(F_ 0)=\{v|\) \(v(x)=\alpha \cdot v+v_ 0\), \(v_ 0\in {\mathbb{R}}\}\) for the Dirichlet integrand \(F_ 0(x,v,v_ x)=| v_ x|^ 2\). However, instead of forming a foliation, the graphs of the functions \(v\in \mu_{\alpha}^{rec}(F)\) might only form a lamination, i.e. a foliation with gaps. The purpose of this paper is to exhibit examples of integrands F such that these gaps really do occur for all \(\alpha\in {\mathbb{R}}^ n\) with \(| \alpha |\) smaller than some arbitrary constant.
Reviewer: M.Adachi

MSC:
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
49J20 Existence theories for optimal control problems involving partial differential equations
35J30 Higher-order elliptic equations
58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
49Q20 Variational problems in a geometric measure-theoretic setting
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