𝑛-Harmonic Mappings Between Annuli The Art of Integrating Free Lagrangians

Source abstract

The central theme of this paper is the variational analysis of homeomorphisms h : X → o n t o Y h \colon \mathbb X \xrightarrow []{{}_{\!\!\mathrm {onto}\!\!}}\mathbb Y between two given domains X , Y ⊂ R n \mathbb X , \mathbb Y \subset \mathbb R^n . We look for the extremal mappings in the Sobolev space W 1 , n ( X , Y ) \mathscr W^{1,n}(\mathbb X,\mathbb Y) which minimize the energy integral $$E h = ∫ X | | D h ( x ) | | n d x . {\mathscr E}_h=\int _{{\mathbb X}} \,|\!|\, Dh(x) \,|\!|\,^n\, \textrm {d}x.$$ Because of the natural connections with quasiconformal mappings this n \,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 n -harmonic mappings are observed. The underlying integration of nonlinear differential forms, called free Lagrangians , becomes truly a work of art.