Harmonic maps between singular spaces I

We discuss regularity questions for harmonic maps from a $n$-dimensionalRiemannian polyhedral complex $X$ to a non-positively curved metric space. Themain theorems assert, assuming Lipschitz regularity of the metric on the domaincomplex, that such maps are locally Hölder continuous with explicit boundsof the Hölder constant and exponent on the energy of the map and thegeometry of the domain and locally Lipschitz continuous away from the $(n-2)$-skeleton of the complex. Moreover, if $x$ is a point on the $k$-skeleton ($k\leq n-2$) we give explicit dependence of the Hölder exponent at a pointnear $x$ on the combinatorial and geometric information of the link of $x$ in$X$ and the link of the $k$-dimensional skeleton in $X$ at $x$.

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Published 1 January 2010