Harmonic and energy-minimal homeomorphisms. (Q2846680)

The paper gives an overview of the existence and nonexistence of \(n\)-harmonic mappings between two domains in \(\mathbb{R}^n\) and some related topics. These results were obtained by the author and his collaborators in recent years. The paper concerns mainly the minimizers of the \(n\)-harmonic energy NEWLINE\[NEWLINE \mathbb{E}[h]=\int_{X}| Dh(x)| ^n\, \text{d}x NEWLINE\]NEWLINE for mappings \(h\: X\to Y\) where \(X\) and \(Y\) are fixed domains in \(\mathbb{R}^n\). It characterizes annuli \(X\) and \(Y\) in the plane for which there is a conformal homeomorphism or harmonic homeomorphism between them. It studies similar problems for general doubly connected domains or in higher dimension. The question of approximability of the Sobolev homeomorphism in the plane by smooth homeomorphisms and its connection with \(p\)-harmonic replacements is also mentioned. This is a very interesting and nicely written survey paper.NEWLINENEWLINEFor the entire collection see [Zbl 1262.35002].