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


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)
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