JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 61, 357-369 (1977) On the Regular Solution of a Quasi-Variational Inequality Connected to a Problem of Stochastic Impulse Control J. L. JOLY,* U. MOSCO,+ AND G. M. TROIANIELLO~ * Universi& de Bordeaux .I, + Istituto Matematico della Universitd di Roma e GNAFA, CNR, and 2 Istituto per le Applicazioni de1 Calcolo de1 CNR e Universitd di Roma Submitted by J. L. Lions INTRODUCTION It has been recently proved, by Bensoussan and Lions (for a list of references, see [S]), that a number of problems of stochastic impulse control can be fitted into the frame of quasi-variational inequaZities (QVI), provided the corresponding solutions are sufficiently smooth. In the present paper (see also [5]) we shall deal with an elliptic QVI in a bounded domain, relative to an operator with constant coefficients and to homogeneous boundary data. The existence of a solution of this QVI has already been proved in a larger context by Bensoussan et al. [l]. By adopting the general framework described in [4], we shall prove the existence of a regular solution. Moreover, we shall show that our solution is the common limit of two monotone sequences of subsohtions and supersolutions (in the terminology of Tartar [12]). As a collateral result, we shall also prove the existence of regular solutions for a class of approximated QVI’s, also with the prospect of numerical utilization. 1 Let G denote a bounded open subset of Rn with a boundary aG of class C2. We define HI(G) as the completion of C’(n) with respect to the norm and H,,l(sZ) as the closure of C,,l(Q) in P(Q); we denote by H-l&?) the dual of H,l(G), and use the symbol (., .) for the duality pairing between elements of 357 Copyright 0 1977 by Academic Press, Inc. All rights of reproduction in any form reserved. ISSN 0022-247X