Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients
Shigeaki Koike Andrzej Świech
Communications on Pure & Applied Analysis 2012, 11(5): 1897-1910 doi: 10.3934/cpaa.2012.11.1897
We establish local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic partial differential equations with unbounded ingredients.
keywords: viscosity solutions Fully nonlinear elliptic PDE local maximum principle.
Remarks on the comparison principle for quasilinear PDE with no zeroth order terms
Shigeaki Koike Takahiro Kosugi
Communications on Pure & Applied Analysis 2015, 14(1): 133-142 doi: 10.3934/cpaa.2015.14.133
A comparison principle for viscosity solutions of second-order quasilinear elliptic partial di erential equations with no zeroth order terms is shown. A di erent transformation from that of Barles and Busca in [3] is adapted to enable us to deal with slightly more general equations.
keywords: comparison principle quasilinear elliptic equation. Viscosity solution
A linear-quadratic control problem with discretionary stopping
Shigeaki Koike Hiroaki Morimoto Shigeru Sakaguchi
Discrete & Continuous Dynamical Systems - B 2007, 8(2): 261-277 doi: 10.3934/dcdsb.2007.8.261
We study the variational inequality for a 1-dimensional linear-quadratic control problem with discretionary stopping. We establish the existence of a unique strong solution via stochastic analysis and the viscosity solution technique. Finally, the optimal policy is shown to exist from the optimality conditions.
keywords: Viscosity solution stopping time linear-quadratic control.
Pursuit-evasion games with state constraints: dynamic programming and discrete-time approximations
Martino Bardi Shigeaki Koike Pierpaolo Soravia
Discrete & Continuous Dynamical Systems - A 2000, 6(2): 361-380 doi: 10.3934/dcds.2000.6.361
In this paper we study the boundary value problem for the Hamilton-Jacobi-Isaacs equation of pursuit-evasion differential games with state constraints. We prove existence of a continuous viscosity solution and a comparison theorem that we apply to establish uniqueness of such a solution and its uniform approximation by solutions of discretized equations.
keywords: comparison principle state-constraint problem Pursuit-evasion game discrete-time approximation. viscosity solution

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