n-Harmonic Mappings Between Annuli: The Art of Integrating Free Lagrangians (Memoirs of the American Mathematical Society)
n-Harmonic Mappings Between Annuli: The Art of Integrating Free Lagrangians (Memoirs of the American Mathematical Society) by Tadeusz Iwaniec
English | 2012 | ISBN: 0821853570 | 105 Pages | PDF | 847.38 KB
English | 2012 | ISBN: 0821853570 | 105 Pages | PDF | 847.38 KB
The central theme of this paper is the variational analysis of homeomorphisms $h: {\mathbb X} \overset{\textnormal{\tiny{onto}}}{\longrightarrow} {\mathbb Y}$ between two given domains ${\mathbb X}, {\mathbb Y} \subset {\mathbb R}^n$. The authors look for the extremal mappings in the Sobolev space ${\mathscr W}^{1,n}({\mathbb X},{\mathbb Y})$ which minimize the energy integral ${\mathscr E}_h=\int_{{\mathbb X}} \,|\!|\, Dh(x) \,|\!|\,^n\, \textrm{d}x$. 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.