JOURNAL OF DIFFERENTIAL EQUATIONS 67, 145-164 (1987) Regularization and Graph Approximation of a Discontinuous Evolution Problem MANUEL D. P. MONTEIRO MARQUES* Centro de Matemcitica e Aplicagdes Fundamentais (CMAF), Av. ProJ Gama Pinto, 2, 1699, Lisboa Codex, Portugal Received May 22, 1984; revised October 28, 1985 1. INTRODUCTION Let H be a real Hilbert space and let I be a compact interval of the real line, say I= [0, T]. Consider a multifunction (set-valued mapping) t + C(t) from I to non-empty closed convex subsets of H. Recall that the Hausdorff distance between two (non-empty closed) subsets A and B of a metric space (E, 6) is defined by with h(A, B)=max{e(A, B), e(B, A)} e(A, B) = sup inf &a, b) = sup dist(a, B). utA bsB fItA We shall assume the existence of real nondecreasing right-continuous functions I(. ) and v(. ) defined on I and such that, for all 0 < s f t < T, e(C(s), C(t)) d r(t) - 4s) resp. We shall refer to these assumptions by saying that C has right-continuous bounded retraction, resp. variation. Classical reasoning shows that there exists an optimal choice among the functions r, resp. v, securing these inequalities; if r and v are so chosen, we write, for s 6 t in I, ret( C; S, t) = r(t) - r(s) * This work was supported by a grant from Funda$o Calouste Gulbenkian (Lisboa), while the author was on leave from the Faculdade de Ci&ncias de Lisboa at the Institut de Mathbmatiques de l’universitt- des Sciences et Techniques du Languedoc, Montpellier. 145 0022-0396187 $3.00 CopyrIght (?) 1987 by Academic Press, Inc. All rights of reproduction in any form reserved. brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by Elsevier - Publisher Connector