Irregular obstacles and quasi-variational inequalities of stochastic impulse control

Frehse, Jens; Mosco, Umberto

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 9 (1982) no. 1, pp. 105-157.

[1] A. Bensoussan - J.L. Lions, Contrôle impulsionnel et contrôle continu. Méthode des I.Q.V. non linéaires, C. R. Acad. Sci. Paris, Série A, 278 (1974), pp. 675-679, 747-751. | Zbl

[2] A. Bensoussan - J.L. Lions, Optimal impulse and continuous control: Method of non linear quasi-variational inequalities, Trudy Mat. Inst. Steklov, 134 (1975), pp. 5-22. | MR | Zbl

[3] A. Bensoussan - J.L. Lions, Sur le controle impulsionnel et les inéquations quasi-variationnelles d'evolution, C. R. Acad. Sci. Paris, Série A, 280 (1975), pp. 1049-1053. | MR | Zbl

[4] A. Bensoussan - J. Frehse - U. Mosco, A stochastic impulse control problem with quadratic growth Hamiltonian and the corresponding quasi variational inequality, J. Reine Angew. Math., to appear. | Zbl

[5] M. Biroli, A De Giorgi-Nash-Moser result for variational inequalities, Boll. Un. Mat. Ital. (5) 16-A (1979), pp. 598-605. | MR | Zbl

[6] M. Biroli, An estimate on convergence of approximation by iterations of a solution to a quasi variational inequality and some consequences on continuous dependence and G-convergence, Ann. Mat. Pura Appl., to appear. | MR | Zbl

[7] L. Caffarelli - A. Friedman, Regularity of the solution of the Q.V.I. for the impulse control problem, Comm. Partial Diff. Eq., 3 (1978), pp. 745-753. Idem II, ibidem. | MR | Zbl

[8] J. Frehse, On the smoothness of solutions of variational inequalities with obstacles, Proc. Semester Partial Diff. Eq., Banach Center, Warszawa 1978.

[9] J. Frehse, On Signorini's problem and variational problems with thin obstacles, Ann. Scuola Norm. Sup. Pisa, 4 (1977), pp. 343-362. | Numdam | MR | Zbl

[10] J. Frehse, On the regularity of solutions to elliptic differential inequalities, J. Reine Angew. Math., to appear. | MR

[11] J. Frehse - U. Mosco, Sur la régularité de certaines inéquations variationnelles et quasi -variationnelles, C. R. Acad. Sci. Paris, Série A, 289 (1979), pp. 627-630. | MR | Zbl

[12] J. Frehse - U. Mosco, Variational inequalities with onesided irregular obstacles, Manuscripta Math., 28 (1979), pp. 219-233. | MR | Zbl

[13] M.G. Garroni - G.M. Troianiello, Some regularity results and a priori estimates for solutions of variational and quasi-variational inequalities, Proc. Conf. Recent methods in nonlinear analysis and Applications, Roma 1978, edited by E. DE GIORGI, E. MAGENES and U. Mosco, Pitagora Ed., Bologna, 1979. | MR | Zbl

[14] B. Hanouzet - J.L. Joly, Méthodes d'ordre dans l'interpretation de certaines inéquations variationnelles, C. R. Acad. Sci. Paris, Série A, 281 (1975), pp. 373-376. | MR | Zbl

[15] B. Hanouzet - J.L. Joly, Convergence uniforme des iterées définissant la solution d'une inequation quasi-variationnelle, C. R. Acad. Sci. Paris, Série A, 286 (1978), pp. 735-738. | MR | Zbl

[16] S. Hildebrandt - K.O. Widman, Some regularity results for quasi-linear elliptic systems of second order, Math. Z., 142 (1975), pp. 67-86. | MR | Zbl

[17] S. Hildebrandt - K.O. Widman, On the Hölder continuity of Weak Solutions of quasi-linear elliptic systems of second order, Ann. Scuola Norm. Sup. Pisa, 4 (1977), pp. 145-178. | Numdam | MR | Zbl

[18] J.L. Joly - U. Mosco, A propos de l'existence et de la régularité des solutions de certaines inéquations quasi-variationnelles, J. Functional Analysis, 34 (1979), pp. 107-197. | MR | Zbl

[19] J.L. Joly - U. Mosco - G.M. Troianiello, On the regular solution of a quasi variational inequality connected to a problem of stochastic impulse control, J. Math. Anal. Appl., 61 (1977), pp. 357-369. | MR | Zbl

[20] Ky Fan, A minimax inequality and applications, in Inequalities III, Shisha ed., Academic Press, New York, 1972, pp. 103-113. | MR | Zbl

[21] O.A. Ladyzhenskaia - N. Ural'Tseva, Linear and quasi-linear elliptic equations, Acad. Press, New York, 1979.

[22] H. Lewy - G. Stampacchia, On the regularity of the solution of a variational inequality, Comm. Pure Appl. Math., 22 (1969), pp. 153-188. | MR | Zbl

[23] H. Lewy - G. Stampacchia, On the smoothness of superharmonies which solve a minimum problem, J. Analyse Math., 23 (1970), pp. 227-236. | MR | Zbl

[24] J.L. Lions - G. Stampacchia, Variational inequatities, Comm. Pure Appl. Math., 20 (1967), pp. 493-519. | MR | Zbl

[25] J.L. Menaldi, Sur le problème de contrôle impulsionnel et l'inéquation quasi variationnelle degenerée associée, C. R. Acad. Sci. Paris, Série A, 284 (1977), pp. 1499-1502. | MR | Zbl

[26] U. Mosco - G.M. Troianiello, On the smoothness of solutions of unilateral Dirichlet problems, Boll. Un. Mat. Ital., 8 (1973), pp. 57-67. | MR | Zbl

[27] U. Mosco, Implicit variational problems and quasi variational inequalities, in Nonlinear operators and calculus of variations, Proc. Bruxelles, 1975, edited by L. WAELBROECK, Lecture Notes in Math., 543, Springer-Verlag, 1976. | MR | Zbl

[28] U. Mosco, Nonlinear quasi-variational inequalities and stochastic impulse control theory, Proc. Conf. Equadiff IV, Praha, 1977, edited by J. FÁBERA, Lecture Notes in Math., 703, Springer-Verlag, 1979. | MR | Zbl

[29] U. Mosco, On some nonlinear quasi variational inequatities and implicit complementarity problems in stochastic control theory, in Variational Inequalities, Proc., edited by R. W. COTTLE, F. GIANNESSI and J. L. LIONS, J. Wiley, 1979. | Zbl

[30] G. Stampacchia, Formes bilinéaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris, Série A, 258 (1964), pp. 4413-4416. | MR | Zbl

[31] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier Grenoble, 15 (1965), pp. 189-258. | Numdam | MR | Zbl

[32] F. Tomi, Variationsprobleme vom Dirichlet-type mit einer Ungteichung als Nebenbedingung, Math. Z., 128 (1972), pp. 43-74. | MR | Zbl

[33] K.O. Widman, Hölder continuity of solutions of elliptic systems, Manuscripta Math., 5 (1971), pp. 299-308. | MR | Zbl

[34] K.O. Widman, The singularity of the Green function for non-uniformly elliptic partial differential equations with discontinuous coefficients, Technical report, Uppsala Univ., 1970.