JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 65, 1lo-125 (1978) On Strong Solutions to Parabolic Unilateral Problems with Obstacle Dependent on Time PIERRE CHARRIER U.E.R. de MathLmatiques et Informatique et Laboratoire associe’ au C.N.R.S. no 226, Universite’ de Bordeaux I, 33405 Talence, France GIOVANNI M. TROIANIELLO Istituto Matematico, Universitci di Roma, 00185 Rome, Italy Submitted by J. L. Lions 1. INTRODUCTION In [16] Lions and Stampacchia proved the existence of weak solutions to vuriutionaZ inequalities (VI) f or p urabolic operators, the constraints assumed to be independent of time; further results can be found in [3, 131. In the case of convex sets depending “regularly” on time, a theorem of existence of weak solutions was proven by Biroli [2]. Finally, Mignot and Puel [17] showed the existence of maximal weak solutions to unilateral problems with obstacles (placed above) depending on time only as measurable functions. We are interested in strong solutions to unilateral problems. The existence of such solutions in the case of obstacles depending “regularly” on time as well as on space variables has been proven by Brezis [4]. Here (see also our previous paper [q) we extend Brezis’ result, by utilizing different techniques. The operator we deal with has essentially bounded measurable coefficients, and the conditions we impose on the lateral boundary of the cylinder are, roughly speaking, of a “mixed” type. As for the obstacle, the major hypotheses we make are expressed in the dual of the space where the test functions lie. Our proof is based on a “dual estimate” for elliptic problems, first proven by Lewy and Stampacchia [12] in the case of boundary conditions of the Dirichlet type. Various extensions of the previous result have been given in [8, 18, 221; here, we apply the formulation of [8] to a “regularized elliptic” problem, then passing to the limit on the perturbating parameter. We prove existence and uniqueness of a strong solution, together with a “parabolic estimate” of the Lewy-Stampacchia type. In particular, this estimate 110 0022-247X/78/0651-01 10$02.OO/O Copyright 0 1978 by Academic Press, Inc. All rights of reproduction in any form reserved.