Stable defects of minimizers of constrained variational principles (Q1110804)

The authors investigate minimizers of certain constrained variational problems the prototype of which arises in the study of liquid crystals. A special case is that of mappings \(u\in H^ 1(\Omega;S^ 2)\), \(\Omega \subset {\mathbb{R}}^3\) a bounded domain, minimizing Dirichlet's integral \(\int_{\Omega}| Du|^2\,dx\). The stationary points in this case are harmonic maps from \(\Omega\) into \(S^2.\) This paper may be regarded as a continuation of the authors' previous work [Commun. Math. Phys. 105, 547--570 (1986; Zbl 0611.35077)]. There they showed regularity of \(u\) near points a where the normalized energy \(E_{r,a}(u)=(1/r)\int_{\Omega}| Du|^2\,dx\) is sufficiently small. Here, they first derive an ``energy density bound'' for minimizers \(u\), i.e., \[ \lim_{r\downarrow 0} \sup E_{r,a}(u)\leq M, \] \(M\) independent of \(u\) and \(a\). Furthermore, an ``interior energy bound'' is proved. In addition, they show higher integrability of \(Du\) which implies that the singular set has dimension less than one. As a consequence certain compactness theorems for minimizers turn out to be true similar to those holding for uniformly bounded families of harmonic functions on the disk. A main ingredient in the proofs are several extension lemmas.