Optimal regularity theorems for variational problems with obstacles (Q5903037)

Starting from the results obtained in our previous papers [see \textit{M. Fuchs}, Analysis 5, 223-238 (1985); Manuscr. Math. 54, 107-119 (1985; Zbl 0587.49011) and \textit{M. Fuchs} and \textit{F. Duzaar}, Math. Z. 191, 585-591 (1986; Zbl 0568.49009)] we here give a complete answer to the regularity question of maps \(u: X\to Y\) beween Riemannian manifolds which minimize energy under a side-condition of the form Im(u)\(\subset \bar M\) for a bounded domain M in the target manifold with smooth boundary \(\partial M\). Here Y is assumed to be embedded in some Euclidean space \({\mathbb{R}}^ N\) in order to define the Sobolev space \(H^ 1(Int X,Y)\) and the subclass \(C:=\{u\in H^ 1(Int X,Y):\) Im(u)\(\subset \bar M\}\). Our theorems are: A local minimizer \(u\in C\) is of class \(C^ 1\) up to a closed singular set S with \({\mathbb{H}}-\dim (S)\leq n-3\); for three dimensions S is discrete. If in addition u minimizes energy in C for fixed smooth boundary values \(\partial X\to \bar M\), then u is \(C^ 1\) in a tubular neighborhood of \(\partial X\). A simple example shows optimality of these statements. The method of proof is based on ideas of \textit{R. Schoen} and \textit{K. Uhlenbeck} [J. Differ. Geom. 17, 307-335, Correction ibid. 19, 329 (1982; Zbl 0521.58021); and ibid. 18, 253-268 (1983; Zbl 0547.58020)]. Concerning the case of (unconstrained) harmonic maps we have to give new arguments to handle the side condition Im(u)\(\subset \bar M\) imposed on the comparison functions.