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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.

##### 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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