Minimizing pseudo-harmonic maps in manifolds
This paper considers the existence, uniqueness and regularity of certain generalized harmonic maps from a bounded domain in \(\mathbb{R}^N\) to a hypersurface \(N\subset\mathbb{R}^{n+1}\) which is the graph over some ellipsoid in \(\mathbb{R}^n\). In particular, \(N\) has a high degree of symmetry, in that when the ellipsoid is given by \((y,y)<1\), \(y\in\mathbb{R}^n\), for a positive quadratic form \((\;,\;)\), the space \(N\) is the graph of a function \(f\) which only depends on \((y,y)\). Given a bounded domain \(\Omega\subset\mathbb{R}^N\) and suitably smooth boundary data \(g:\partial\Omega\to N\) the author considers \(H^1\)-solutions to a version of the harmonic map equations for maps \(u:\Omega\to N\) with boundary \(g\): these harmonic maps are only presumed to be weak solutions in the sense of distribution theory and are critical points of an appropriate energy functional \(E(u)\). NEWLINENEWLINENEWLINEThere are three main results of the paper, all of which seem to require certain conditions on the first three derivatives of the function \(f\) (whose graph determines \(N\)). First, the minimizer of \(E(u)\) is unique in \(H^1\), and any harmonic \(u\) whose image lies in a compact subset of \(N\) is the minimizer. Second, any harmonic \(u\) with image in a compact subset of \(N\) is \(C^2\). Third, when \(n=N=2\), \(\Omega\) is the unit disc and the boundary data is a convex Jordan curve in a horizontal plane (i.e. parallel to the plane over which the graph \(N\) lives) then \(u\) is a diffeomorphism.
scientific article; zbMATH DE number 1605718