A new proof of regularity of weak solutions of the <Emphasis Type="Italic">H</Emphasis>-surface equation

Abstract. We give a new proof of a theorem of Bethuel, asserting that arbitrary weak solutions of the H-surface system are locally H?lder continuous provided that H 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 , this time assuming in addition that decays at infinity like . Received: 10 May 2001 / Accepted: 7 June 2001 / Published online: 18 January 2002 The author gratefully acknowledges the generous support of Alexander von Humboldt Foundation, and the hospitality of Mathematisches Institut der Universit?t Bonn, where this research has been carried out. In particular, many thanks are due to Professor Stefan Hildebrandt.