Set-Valued Analysis 5: 57–72, 1997. 57 c  1997 Kluwer Academic Publishers. Printed in the Netherlands. BV Solutions to Evolution Problems with Time-Dependent Domains M. KUNZE 1 and M. D. P. MONTEIRO MARQUES 2 1 Mathematisches Institut der Universit¨ at K¨ oln, Weyertal 86, D - 50931 K¨ oln, Germany 2 CMAF and Faculdade de Ciˆ encias da Universidade de Lisboa, Av. Prof. Gama Pinto 2, P -1699 Lisboa, Portugal (Received: 20 February 1996; in final form: 8 July 1996) Abstract. In a Hilbert space H we consider evolution problems −du(t) ∈ A(t)u(t) on some interval [0,T ], where every A(t): D(A(t)) → 2 H is a maximal monotone operator, and the correspondence t → A(t) is – in a suitable sense – of bounded variation or absolutely continuous. Mathematics Subject Classifications (1991). 34A60, 35K22. Key words: maximal monotone operators, pseudo-distance, absolutely continuous, bounded vari- ation. 1. Introduction and Main Results In this paper, we first study evolutionary problems of the type − du dr (t) ∈ A(t)u(t) dr-a.e. in [0,T ], (1) where A(t): D(A(t)) ⊂ H → 2 H is a maximal monotone operator (mmop) in a Hilbert space H for every t ∈ [0,T ], and the dependence t → A(t) is – in some sense – of bounded variation (b.v.). To be more precise, we assume that (H1) there exists a function r: [0,T ] → [0, ∞[ which is right-continuous on [0,T [ and nondecreasing with r(T ) < ∞ such that dis(A(t),A(s))  dr(]s,t]) = r(t) − r(s) for 0  s  t  T, (2) where dis(·, ·) is the pseudo-distance between mmops introduced in [9], and given through dis(A,B) = sup  〈y − ¯ y, ¯ x − x〉 1 + |y| + | ¯ y| : x ∈ D(A),y ∈ Ax, ¯ x ∈ D(B), ¯ y ∈ B ¯ x  (3) for mmops A and B in H with domains D(A), resp. D(B). VTEX(LK) PIPS No.:120911 MATHKAP SVAN286.tex; 3/04/1997; 15:44; v.5; p.1