On the Size of the Singular Set of Minimizing Harmonic Maps

We consider minimizing harmonic maps u u from Ω ⊂ R n Rⁿ into a closed Riemannian manifold N N and prove: 1. an extension to n ≥ 4 n 4 of Almgren and Lieb’s linear law. That is, if the fundamental group of the target manifold N N is finite, we have \ H n − 3 (sing ⁡ u) ≤ C ∫ ∂ Ω | ∇ T u | n − 1 d H n − 1 ; H^n-3 (sing u) C | T u|^n-1 \, d H^n-1; \ 2. an extension of Hardt and Lin’s stability theorem. Namely, assuming that the target manifold is N = S 2 N= S² we obtain that the singular set of u u is stable under small W 1, n − 1 W^1, n-1 -perturbations of the boundary data. In dimension n = 3 n=3 both results are shown to hold with weaker hypotheses, i. e. , only assuming that the trace of our map lies in the fractional space W s, p W^s, p with s ∈ (1 2, 1 ] s (12, 1] and p ∈ [ 2, ∞) p [2, ) satisfying s p ≥ 2 sp 2. We also discuss sharpness.