Energy minimizing maps to piecewise uniformly regular Lipschitz manifolds
This paper generalizes the regularity theory for energy minimizing harmonic maps [see \textit{R. Schoen} and \textit{K. Uhlenbeck}, J. Differ. Geom. 17, 307-335 (1982; Zbl 0521.58021)] to the case of maps from a domain in \(R^m\) to piecewise uniformly regular Lipschitz manifolds -- a concept including both \(C^1\) manifolds and polyhedral submanifolds. There may be, in general, no Euler-Lagrange equation for energy minimizers. However, most of the partial regularity proof for minimizing harmonic maps [in the form simplified by \textit{S. Luckhaus}, Indiana Univ. Math. J. 37, 349-368 (1988; Zbl 0641.58012)] carries over to the case considered here, except, due to the lack of a differential equation, higher differentiability. An additional ingredient which is needed here is Hölder continuity of energy minimizing maps to any tangent cone of the target Lipschitz manifolds. In order to combine this with the usual blowup arguments, one needs a monotonicity formula, an inductive description of the structure of tangent cones, and some uniform approximability of the target manifold by its tangent cones. The result is Hölder continuity of energy minimizers outside a closed singular set of codimension \(\geq 3\). For a polyhedral target, besides partial regularity, it is also proved that the preimage of the \((k-2)\)-skeleton under an energy minimizing map has codimension \(\geq 1\) if the map is not constant.
piecewise uniformly regular Lipschitz manifolds
Hölder continuity of energy minimizing maps
scientific article; zbMATH DE number 1833136