JOURNAL OF DIFFERENTIAL EQUATIONS 94, 83-94 (1991) C’s’-Regularity of Constrained Area Minimizing Hypersurfaces PETERSTERNBERG* AND WILLIAM P. ZIEMER' Mathematics Department, Indiana University. Bloomington, lndiuna 47405 AND GRAHAM WILLIAMS~ Mathematics Department, Wollongong University, Australia Received February 26, 1990 In [SWZ] we considered the constrained least gradient problem inf iI lVu[ dx: u E Cog’, lVul d 1 a.e.,24 = g on asz , R 1 where Q c R" is an open set and g is a continuous function on XJ satis- fying the Lipschitz condition )g(p) - g(q)\ < 1 p - qj for all p, q E aQ. This study was motivated by work of Kohn and Strang [KoS] who showed that (1) arises as the relaxation of a nonconvex variational problem in optimal design. The main purpose of [SWZ] was to show that the solution u to (1) can be explicitly constructed by equating the set {U 3 r} with the solution to the obstacle problem inf(P(E,Q):G;Z E=,L,.i?n(M)'=@), (2) where P(E, Q) denotes the perimeter of E in 0 and where L and A4 are closed sets that depend on t and satisfy an interior ball condition of radius R. That is, for each x E L, we assumethere is a ball B c L of radius r >/ R such that x E B, and similarly for M. We also assume that L n (AI)'= 0. This naturally led us to the question of regularity of solutions of (2). The * Research supported in part by a grant from the National Science Foundation and an Indiana University Summer Faculty Fellowship. + Research supported in part by a grant from the National Science Foundation. : Research conducted while visiting Indiana University. 83 0022-0396191 $3.00 Copyright Q 1991 by Academic Press, Inc All rights of reproduction m any form reserved.