A partial regularity theorem for harmonic maps at a free boundary

Abstract

For a Riemannian manifold M with boundary, a Riemannian manifold N, an open subset Σ≠∅ of ∂M and a submanifold S of N, we prove that every map u:M→N of class H^1,2 which is energy minimizing with respect to the free boundary condition u(Σ)⊂S is regular on an open subset Reg(u) of M∪Σ whose complement Sing(u)=(M∪Σ)\Reg(u) has vanishing m−2 dimensional measure. The result is new for the singular set Σ∩Sing(u) contained in the free boundary Σ. There are no restrictions on the image of u in N and no dimensional restrictions on M, N and S.

The proof is based on a method introduced by R. Schoen and K. Uhlenbeck in the interior partial regularity theory for harmonic mappings, a novel partial reflection construction ũ extending u and a precise control of the local averages u_r,a, ũ_r,a on balls B_r,a with radius r>0 and center a∈Σ in terms of the small normalized energy ε of u on B_r,a. The crucial estim ate is dist(u_r,a,S) $\lesssim$ ε^1/2 and dist(ũ_r,a,S) $\lesssim$ ε^2/3 as ε↓0.