Skip to main content
SpringerLink
Log in

Über Mosers regularisiertes Variationsproblem für minimale Blätterungen desn-dimensionalen Torus

  • Original Papers
  • Published:
Zeitschrift für angewandte Mathematik und Physik ZAMP Aims and scope Submit manuscript

Abstract

We consider regular and Cantor-like minimal foliations of the (n+1)-dimensional TorusT n+1 whose leaves minimize a given variational integral. Each leaf of such a generalized foliation lies in the universal coveringR n+1 within a finite distance to the affine leaves (z, αx+β) of fixed α εR n. We show that the conjugation-functionU α (x,θ), mapping the affine leaves (x, αx+β) into the leaves(x,U α (x,xα+β)) of the generalized foliation, is itself a minimal solution of an extended degenerate variational problem onT n +1. If α ∈R n/Q n the functionU α is characterized in a unique way as (discontinuous) limit of the minimal solutions of the corresponding regularized problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literatur

  1. Bangert, V.,A uniqueness theorem for Z n -periodic variational problems. Comment. Math. Helv.62, 511–531 (1987).

    Google Scholar 

  2. Bangert, V.,On minimal laminations of the torus. Ann. Inst. Henri Poincaré (Analyse non linéaire)6, 95–138 (1987).

    Google Scholar 

  3. Denzler, J.,Mather sets for plane Hamiltonian systems. J. Appl. Math. Phys. (ZAMP)38, 791–812 (1987).

    Google Scholar 

  4. Mather, J.,Existence of quasi-periodic orbits for twist homeomorphisms of the annulus. Topology21, 457–467 (1982).

    Google Scholar 

  5. Moser, J.,Minimal solutions of variational problems on a torus. Ann. Inst. Henri Poincaré (Analyse non linéaire)3, 229–272 (1986).

    Google Scholar 

  6. Moser, J.,Minimal Foliation on a Torus. In:Topics in Calculus of Variations (Montecantini Terme 1987), ed. M. Giaquinta, Springer Lect. Notes in Math.1365, 62–99 (1989).

  7. Rudin, W.,Real and Complex Analysis, McGraw-Hill, New York 1966.

    Google Scholar 

  8. Senn, W.,Strikte Konvexität für Variationsprobleme auf dem n-dimensionalen Torus. Eingereicht beiManuscriptu mathematica.

  9. Weyl, H.,Über die Gleichverteilung mod. Eins. Math. Ann.77, 313–352 (1916).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut der Universität Bern, Sidlerstrasse 5, 3006, Bern

    W. Senn

Authors
  1. W. Senn
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Senn, W. Über Mosers regularisiertes Variationsproblem für minimale Blätterungen desn-dimensionalen Torus. Z. angew. Math. Phys. 42, 527–546 (1991). https://doi.org/10.1007/BF00946175

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00946175

Search

Navigation