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Boundary regularity of weakly harmonic maps from surfaces. (English) Zbl 0785.53048

The works of Morrey and Schoen-Uhlenbeck provide a boundary regularity theory for minimizing harmonic maps from surfaces. The main result of the paper completes the theory by removing minimality and requiring only weak harmonicity. More precisely, the author proves that a weakly harmonic map \(u \in H'(D,N)\) of the unit disk into a compact Riemannian manifold \(N\) is continuous (resp. smooth) provided that the boundary map \(u| \partial D\) is continuous (resp. smooth). Among the main technical tools, the author uses a local maximum principle by improving Helein’s interior regularity and the Courant-Lebesgue lemma.
Reviewer: G.Tóth (Camden)

MSC:

53C40 Global submanifolds
58E20 Harmonic maps, etc.
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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