A new proof of regularity of weak solutions of theH-surface equation

We give a new proof of a theorem of Bethuel, asserting that arbitrary weak solutions u ∈ W (IB, IR) of the H-surface system ∆u = 2H(u)ux1 ∧ ux2 are locally Hölder continuous provided thatH is a bounded Lipschitz function. Contrary to Bethuel’s, our proof completely omits Lorentz spaces. Estimates below natural exponents of integrability are used instead. (The same method yields a new proof of Hélein’s theorem on regularity of harmonic maps from surfaces into arbitrary compact Riemannian manifolds.) We also prove that weak solutions with continuous trace are continuous up to the boundary, and give an extension of these results to the equation of hypersurfaces of prescribed mean curvature in IR, this time assuming in addition that |∇H(y)| decays at infinity like |y|−1.