Energy minimizing maps to piecewise uniformly regular Lipschitz manifolds
type: Item
instance of: scholarly article
title: Energy minimizing maps to piecewise uniformly regular Lipschitz manifolds
zbMATH Open document ID: 1021.58012
author: Changyou Wang
published in: Communications in Analysis and Geometry
publication date: 2003-02-04T00:00:00Z
review text: 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.
reviewed by: Andreas Gastel
Mathematics Subject Classification ID: 58E20
zbMATH DE Number: 1833136
zbMATH Keywords: polyhedral manifolds
zbMATH Keywords: partial regularity
zbMATH Keywords: tangent cones
zbMATH Keywords: energy minimizing harmonic maps
zbMATH Keywords: piecewise uniformly regular Lipschitz manifolds
zbMATH Keywords: \(C^1\) manifolds
zbMATH Keywords: Hölder continuity of energy minimizing maps
MaRDI profile type: Publication
DOI: 10.4310/CAG.2001.V9.N4.A1
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description: scientific article; zbMATH DE number 1833136