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$n$-Harmonic Mappings Between Annuli The Art of Integrating Free Lagrangians
About this Title
Tadeusz Iwaniec, Department of Mathematics, Syracuse University, Syracuse, New York 13244 and Department of Mathematics and Statistics, University of Helsinki, Finland and Jani Onninen, Department of Mathematics, Syracuse University, Syracuse, New York 13244
Publication: Memoirs of the American Mathematical Society
Publication Year:
2012; Volume 218, Number 1023
ISBNs: 978-0-8218-5357-3 (print); 978-0-8218-9008-0 (online)
DOI: https://doi.org/10.1090/S0065-9266-2011-00640-4
Published electronically: September 19, 2011
Keywords: $n$-Harmonics,
Extremal problems,
Quasiconformal mappings,
Variational integrals
MSC: Primary 30C65, 30C75, 35J20
Table of Contents
Chapters
- Preface
- 1. Introduction and Overview
1. Principal Radial $n$-Harmonics
- 2. Nonexistence of $n$-Harmonic Homeomorphisms
- 3. Generalized $n$-Harmonic Mappings
- 4. Notation
- 5. Radial $n$-Harmonics
- 6. Vector Calculus on Annuli
- 7. Free Lagrangians
- 8. Some Estimates of Free Lagrangians
- 9. Proof of Theorem
2. The $n$-Harmonic Energy
- 10. Harmonic Mappings between Planar Annuli, Proof of Theorem
- 11. Contracting Pair, $\mbox {Mod}\, {\mathbb A}^{\! \ast } \leqslant \mbox {Mod}\, {\mathbb A}$
- 12. Expanding Pair, $\mbox {Mod}\, {\mathbb A}^{\! \ast } > \mbox {Mod}\, {\mathbb A}$
- 13. The Uniqueness
- 14. Above the Upper Nitsche Bound, $n \geqslant 4$
- 15. Quasiconformal Mappings between Annuli
Abstract
The central theme of this paper is the variational analysis of homeomorphisms $h \colon \mathbb X \xrightarrow []{{}_{\!\!\mathrm {onto}\!\!}}\mathbb Y$ between two given domains $\mathbb X , \mathbb Y \subset \mathbb R^n$. We look for the extremal mappings in the Sobolev space $\mathscr W^{1,n}(\mathbb X,\mathbb Y)$ which minimize the energy integral \[ \mathscr E_{h}=∫_{\mathbb X} || Dh(x) || ^{n} dx. \] Because of the natural connections with quasiconformal mappings this $\,n$-harmonic alternative to the classical Dirichlet integral (for planar domains) has drawn the attention of researchers in Geometric Function Theory. Explicit analysis is made here for a pair of concentric spherical annuli where many unexpected phenomena about minimal $n$-harmonic mappings are observed. The underlying integration of nonlinear differential forms, called free Lagrangians, becomes truly a work of art.- Stuart S. Antman, Fundamental mathematical problems in the theory of nonlinear elasticity, Numerical solution of partial differential equations, III (Proc. Third Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975) Academic Press, New York, 1976, pp. 35–54. MR 0468503
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