A partial regularity theorem for harmonic maps at a free boundary

Article type: Research Article

Authors: Duzaar, Frank | Steffen, Klaus

Affiliations: Mathematisches Institut, Universität Düsseldorf, D-4000 Düsseldorf 1, Fed. Rep. Germany

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 H1,2 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 ur,a, ũr,a on balls Br,a with radius r>0 and center a∈Σ in terms of the small normalized energy ε of u on Br,a. The crucial estim ate is dist(ur,a,S)$\lesssim$ε1/2 and dist(ũr,a,S)$\lesssim$ε2/3 as ε↓0.

DOI: 10.3233/ASY-1989-2403

Journal: Asymptotic Analysis, vol. 2, no. 4, pp. 299-343, 1989