Math. Z. 198, 261-275 (1988) Mathematische Zeitschrift 9 Springer-Verlag 1988 Regularity of a Minimal Surface at its Free Boundary Rugang Ye* Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA 1. Introduction In this paper we shall consider a stationary minimal surface X whose free or at least partially free boundary lies on a given supporting surface S. It is well-known that, if the supporting surface S has no boundary, then X possesses the same smoothness as S does. If, on the other hand, the boundary of S is nonvoid, one can at most expect C t, 1/2 regularity of X at its free boundary, c.f. [1], [5], [6]. This C 1" 1/2 boundary regularity was proved in the paper [6] by S. Hildebrandt and J.C.C. Nitsche and in the paper [5] by M. Griiter, S. Hildebrandt and J.C.C. Nitsche under the condition that the supporting surface is of class C 3'" for some ae(0, 1). Now combining the ideas of D. Kinderlehrer [7] and M. Giaquinta, E. Giusti [4] and modifying the argument in [6] concerning the conformality relation, we are able to show the C 1,1/2 regularity under a C2"-assumption, which, in turn, seems to be the weakest, even if the minimal surface under consideration are minima of the area. We also treat the regularity problem of general stationary surfaces under the C 1,1_condition and of minima of the Dirichlet integral under a C 1,,_condition (thus without conformality assumption). 2. Preparations We shall use the following notations: B={lwJ<l, v>0}, I={Iwl<l, v=0}, Sr(wo)={IW-Wol<r}, Br(wo)=S~(wo)c~B, Ir(wo)=S~(wo)C~I, Za={weB I Iwl<l-d}, de(0, 1), where w=(u,v) denotes a point in the complex plane C. By the so-called supporting surfaces we mean regular surfaces (embedded 2-submanifolds) in N3, possibly with boundary' serving as constraint in our * Research done while author was a member of Sonderforschungsbereich 72, lnstitut ffir Angewandte Mathematik, Bonn.