Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0678.58014
Bangert, V.
On minimal laminations of the torus. (English)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 6, No.2, 95-138 (1989). ISSN 0294-1449

Consider the problem $\int F(x,u(x),u\sb x(x))dx\to \min$ where $u: {\bbfR}\sp n\to {\bbfR}$, $F: {\bbfR}\sp n\times {\bbfR}\times {\bbfR}\sp n\to {\bbfR}$ is periodic in $(x,u)\in {\bbfR}\sp{n+1}$ and uniformly convex in $u\sb x\in {\bbfR}\sp n$. Non-selfintersecting minimizers u are investigated, i.e. the hypersurface graph $(u)\subseteq {\bbfR}\sp{n+1}$ has no selfinteractions when projected into $T\sp{n+1}$, where $T\sp{n+1}$ denotes the torus ${\bbfR}\sp{n+1}/{\bbfZ}\sp{n+1}$. There exists a ``rotation vector'' $\alpha =\alpha (u)\in {\bbfR}\sp n$ for such u so that $u(x)-\alpha$ is bounded uniformly for all $x\in {\bbfR}\sp n$. The structure of the set ${\cal M}\sb{\alpha}={\cal M}\sb{\alpha}(F)$ of non-selfintersecting F-minimal solutions with fixed rotation vector $\alpha$ is determined for rationally dependent ${\bar \alpha}=(-\alpha,1)$. These investigations are primarily topological. $u\in {\cal M}\sb{\alpha}$ are classified by secondary invariants. The proved uniqueness results mean that the graphs of functions in ${\cal M}\sb{\alpha}$ with the same secondary invariants do not intersect. \par Using these results and the results by {\it J. Moser} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3, 229-272 (1986; Zbl 0609.49029)], the existence of minimal solutions $u\in {\cal M}\sb{\alpha}$ with prescribed secondary invariants is proved, particularly the existence of secondary laminations in the gaps between the functions in ${\cal M}\sb{\alpha}$ with maximal periodicity. Further, two open problems are mentioned.
[L.Bakule]

MSC 2000:: *58E15 Appl. of variational methods to extremal problems in sev.variables
37C85 Dynamics of group actions other than $\bbfZ$ and $\bbfR$, etc.
49Q20 Variational problems in geometric measure-theoretic setting

Keywords: $Z\sp n$-periodic variational problem; minimizing solutions; laminations

Citations: Zbl 0609.49029

Cited in: Zbl 1234.35005 Zbl 1169.35323 Zbl 1187.35066 Zbl 1229.35047 Zbl 1149.35341

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]