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A quasilinear elliptic equation in ℝN

Published online by Cambridge University Press:  14 November 2011

O. Alvarez
O. Alvarez
Affiliation:
Analyse et Modèles Stochastiques (URA CNRS 1378), Université de Rouen, 76821 Mont Saint-Aignan Cedex, France e-mail: Olivier.Alvarez@univ-rouen.fr
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Abstract

A quasilinear elliptic equation in ℝN of Hamilton-Jacobi-Bellman type is studied. An optimal criterion for uniqueness which involves only a lower bound on the functions is given. The unique solution in this class is identified as the value function of the associated stochastic control problem.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 5 , 1996 , pp. 911 - 921
Copyright
Copyright © Royal Society of Edinburgh 1996

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References

1Alvarez, O.. Bounded from below solutions of Hamilton-Jacobi equations. Differential Integral Equations (to appear).Google Scholar
2Alvarez, O., Lasry, J. M. and Lions, P. L.. Convex viscosity solutions and state constraints (preprint).Google Scholar
3Brezis, H.. Semilinear equations in ℝN without condition at infinity. Appl. Math. Optim. 12 (1984), 271–82.CrossRefGoogle Scholar
4Crandall, M. G. and Lions, P. L.. Remarks on the existence and uniqueness of unbounded viscosity solutions of Hamilton-Jacobi equations. Illinois J. Math. 31 (1987), 665–88.CrossRefGoogle Scholar
5Crandall, M. G., Newcomb, R. T. and Tomita, Y.. Existence and uniqueness for viscosity solutions ofdegenerate quasilinear elliptic equations in ℝN. Appl. Anal. 34 (1989), 1–23.CrossRefGoogle Scholar
6Fleming, W. H. and Soner, H. M.. Controlled Markov Processes and Viscosity Solutions (New York: Springer, 1993).Google Scholar
7Gilbarg, D. and Trudinger, N. S.. Elliptic Partial Differential Equations of Second Order, 2nd edn (New York: Springer, 1983).Google Scholar
8Krylov, N. V.. Controlled Diffusion Processes (New York: Springer, 1980).CrossRefGoogle Scholar
9Krylov, N. V.. On controlled diffusion processes with unbounded coefficients. Math. USSR Izvestija 19 (1982), 41–64.CrossRefGoogle Scholar
10Lasry, J. M. and Lions, P. L.. Nonlinear elliptic equations with singular boundary conditions andstochastic control with state constraints. Math. Ann. 283 (1989), 583–630.CrossRefGoogle Scholar
11Lions, P. L.. Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Part 1: The dynamic programming principle and applications. Part 2: Viscosity solutions and uniqueness. Comm. Partial Differential Equations 8 (1983), 1101–74; 1229–76.CrossRefGoogle Scholar
12Lions, P. L.. Quelques remarques sur les problemes elliptiques quasilineaires du second ordre. J. Anal. Math. 45 (1985), 234–54.CrossRefGoogle Scholar