## Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints. I: The model problem.(English)Zbl 0688.49026

This paper deals with nonlinear elliptic equations of the form $$-\Delta u+| \nabla u|^ p+\lambda u=f$$, in $$\Omega$$, where $$p>1$$, $$\lambda >0$$ and f is smooth in $$\Omega$$. The authors look for solutions $$u\in C^ 2(\Omega)$$ which either satisfy $$u(x)\to +\infty$$ or $$\partial u(x)/\partial n\to +\infty$$, as dist(x,$$\partial \Omega)\to 0.$$
Singular problems of this form arise if u represents the value function of optimal feedback control problems with infinite horizon for stochastic differential equations. The authors give a thorough treatment of this problem, including both subquadratic and superquadratic Hamiltonians, the viscosity approach, and the ergodic problem (for $$\lambda \to 0+)$$. In the final section of this important paper, the theory is applied to construct the unique optimal markovian control for state constrained optimal stochastic control problems.
Reviewer: J.Sprekels

### MSC:

 49K45 Optimality conditions for problems involving randomness 93E20 Optimal stochastic control 49L20 Dynamic programming in optimal control and differential games 35J55 Systems of elliptic equations, boundary value problems (MSC2000)
Full Text:

### References:

 [1] Amann, H., Crandall, M.G.: On some existence theorems for semilinear elliptic equations. Ind. Univ. Math. J27, 779-790 (1978) · Zbl 0391.35030 [2] Bensoussan, A., Lions, J.L.: Applications des inéquations variationnelles on contrôle stochastique. Paris: Dunod 1978 · Zbl 0411.49002 [3] Bensoussan, A., Lions, J.L.: Contrôle impulsionnel et inéquations quasi-variationnelles. Paris: Dumod 1982 · Zbl 0491.93002 [4] Bombieri, E., De Giorgi, E., Miranda, M.: Una maggiorazioni a priori relativa alle ipersuperfici minimali non parametriche. Arch. Rat. Mech. Anal.32, 255-267 (1969) · Zbl 0184.32803 [5] Bony, J.M.: Principe du maximum dans les espaces de Sobolev. Cr. Acad. Sci. Paris265, 233-236 (1967) [6] Brézis, H.: Semilinear equations in ? n without condition ar infinity. Appl. Math. Opt.12, 271-282 (1984) · Zbl 0562.35035 [7] Crandall, M.G., Lions, P.L.: Remarks on unbounded viscosity solutions of Hamilton-Jacobiequations. Ill. J. Math.31, 665-688 (1987) · Zbl 0678.35009 [8] Fleming, W.H., Rishel, R.: Deterministic and stochastic optimal control. Berlin Heidelberg New York: Springer 1975 · Zbl 0323.49001 [9] Holland, C.J.: A new energy characterization of the smallest eigenvalue of the Schrödinger equation. Comm. Pure Appl. Math.30, 755-765 (1977) · Zbl 0358.35060 [10] Holland, C.J.: A minimum principle for the principal eigenvalue for second-order linear elliptic equations with natural boundary conditions. Comm. Pure Appl. Math.31, 509-519 (1978) · Zbl 0388.35053 [11] Krylov, N.V.: Controlled diffusion processes. Berlin Heidelberg New York: Springer 1980 · Zbl 0436.93055 [12] Krylov, N.V.: Nonlinear second-order elliptic and parabolic equations. Moscow: Nauk 1985 (in Russian) · Zbl 0586.35002 [13] Ladyshenskaya, O., Ural’tseva, N.: Local estimates for gradients of solutions of nonuniformly elliptic and parabolic equations. Comm. Pure Appl. Math.23, 677-703 (1970) · Zbl 0193.07202 [14] Laetsch, T.: On the number of solutions of boundary value problems with convex nonlinearities. J. Math. Anal. Appl.35, 389-404 (1971) · Zbl 0213.38002 [15] Lasry, J.M., Lions, P.L.: In preparation [16] Lions, P.L.: Resolution de problèmes elliptiques quasilinéaires. Arch. Rat. Mech. Anat.74, 336-353 (1980) · Zbl 0449.35036 [17] Lions, P.L.: On the Hamilton-Jacobi-Bellman equations. Acta Applicandae1, 17-41 (1983) · Zbl 0594.93069 [18] Lions, P.L: Some recent results in the optimal control of diffusion processes. In: Stochastic Analysis, Proceedings in the Taniguchi International Symposium on Stochastic Analysis, Katata, Kyoto (1982). Tokyo: Kinokuniya 1984 [19] Lions, P.L.: Quelques remarques sur les problèmes elliptiques quasillineaires du second ordre. J. Anal. Math.45, 234-254 (1985) · Zbl 0614.35034 [20] Lions, P.L.: Viscosity solutions of Hamilton-Jacobi equations and boundary conditions. Preprint · Zbl 0595.35023 [21] Lions, P.L.: Generalized solutions of Hamilton-Jacobi equations. London: Pitman 1982 · Zbl 0497.35001 [22] Lions, P.L.: Neumann type boundary conditions for Hamilton-Jacobi equations. Duke Math. J.52, 793-820 (1985) · Zbl 0599.35025 [23] Lions, P.L.: A remark on Bony maximum principle. Proc. Am. Math. Soc.88, 503-508 (1983) · Zbl 0525.35028 [24] Lions, P.L., Trudinger, N.S.: Linear oblique derivative problems for the uniformly elliptic Hamilton-Jacobi-Bellman equations. Math. Z.191, 4-15 (1985) · Zbl 0593.35046 [25] Lions, P.L., Trudinger, N.S.: In preparation [26] Simon, L.: Interior gradient bounds for nonuniformly elliptic equations. Ind. Univ. Math. J.25, 821-855 (1976) · Zbl 0346.35016 [27] Simon, L.: Boundary regularity for solutions of the nonparametric least area problem. Ann. Math.103, 429-455 (1976) · Zbl 0335.49031 [28] Simon, L.: Global estimates of Hölder continuity for a class of divergence form equations. Arch. Rat. Mech. Anal.56, 223-272 (1974) · Zbl 0295.35027 [29] Soner, M.H.: Optimal control with state-space constraints. I. SIAM J. Control Optim. (1986) · Zbl 0597.49023 [30] Urbas, J.I.E.: Elliptic equations of Monge-Ampere type. Thesis, Univ. of Canberra, Australla, 1984
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.