Some remarks on quasi-variational inequalities and the associated impulsive control problem

B. Perthame

doi:10.1016/s0294-1449(16)30404-8

JournalsaihpcVol. 2, No. 3pp. 237–260

Some remarks on quasi-variational inequalities and the associated impulsive control problem

B. Perthame
E. N. S., 45, rue d’Ulm, 75230 Paris, Cedex 05

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Abstract

We study Quasi-Variational Inequalities:

{Max (A u - f, u - M u) = 0, u ∣_{\partial Ω} = φ, (1)

where

M u = k + x + ξ \in \overset{ˉ}{Ω} ξ ≧ 0 in f {c_{0} (ξ) + u (x + ξ)} . (2)

In general, (1) has no solution, we prove here that (1) has a unique maximum subsolution that we caracterize. Then we compare the implicit obstacle (2) and the obstacle:

M_{+} u = k + x + ξ \in Ω ξ ≧ 0 in f ess {c_{0} (ξ) + u (x + ξ)}

and we finally show that, under general assumptions, the solution of (1) is Holder continuous.

Résumé

Nous étudions les Inéquations Quasi-Variationnelles :

{Max (A u - f, u - M u) = 0 u ∣_{\partial Ω} = φ, dans Ω (1)

où

M u = k + x + ξ \in \overset{ˉ}{Ω} ξ ≧ 0 in f {c_{0} (ξ) + u (x + ξ)} . (2)

En général (1) n’a pas de solution, nous montrons ici que (1) admet une unique sous-solution maximale que nous caractérisons. Nous comparons ensuite l’obstacle implicite (2) et l’obstacle :

M_{+} u = k + x + ξ \in Ω ξ ≧ 0 in f ess {c_{0} (ξ) + u (x + ξ)}

et nous finissons par montrer que, sous des hypothèses générales, la solution de (1) est hölderienne.

Cite this article

B. Perthame, Some remarks on quasi-variational inequalities and the associated impulsive control problem. Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 3, pp. 237–260

DOI 10.1016/S0294-1449(16)30404-8