A Maximum Principle for Generalizations of Harmonic Maps in Hermitian, Affine, Weyl, and Finsler Geometry

Abstract

In this note we prove that the maximum principle of Jäger–Kaul for harmonic maps holds for a more general class of maps, $V$-harmonic maps. This includes Hermitian harmonic maps (Jost and Yau, Acta Math 170:221–254, 1993), Weyl harmonic maps (Kokarev, Proc Lond Math Soc 99:168–194, 2009), affine harmonic maps Jost and Simsir (Analysis (Munich) 29:185–197, 2009), and Finsler maps from a Finsler manifold into a Riemannian manifold. With this maximum principle we establish the existence of $V$-harmonic maps into regular balls.

Appendix

For the convenience of the reader, we recall the extended Sobolev inequality.

Theorem A

(cf. [7], p. 27, Theorem 10.1.) Let $\Omega$ be a bounded domain in ${\mathbb {R}}^m$ with $\partial \Omega \in C^k$, and let $u$ be any function in $W^{k,r}(\Omega )\cap L^q(\Omega ),\,1\le r, q\le \infty$. For any integer $j,\,0\le j <k$, and for any number $a$ in the interval $j/k\le a \le 1$, set

$$\begin{aligned} \frac{1}{p}=\frac{j}{m}+a\left( \frac{1}{r}-\frac{k}{m}\right) +(1-a)\frac{1}{q}. \end{aligned}$$

If $k-j-m/r$ is not a nonnegative integer, then

$$\begin{aligned} \Vert D^ju\Vert ^{0,p}_\Omega \le C (\Vert u\Vert ^{k,r}_\Omega )a(\Vert u\Vert ^{0,q}_\Omega )^{1-a}. \end{aligned}$$

(46)

If $k-j-m/r$ is a nonnegative integer, then (46) holds for $a=j/k$. The constant $C$ depends only on $\Omega , r, q, k, j, a$.