User’s guide to viscosity solutions of second order partial differential equations

By M. Crandall, Hitoshi Ishii, Pierre-Louis Lions. 1992

The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a self-contained exposition of the basic theory of viscosity solutions

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Citations in the corpus

Travelling fronts in asymmetric nonlocal reaction diffusion equations: The bistable and ignition cases (2007, unknown): “We present and develop here an approach briefly discussed by Chen [10] which is inspired by the viscosity solution theory introduced by Crandall - Lions [3, 20, 19].”
A Visit with the ∞-Laplace Equation (2008, methodology): “This is consistent with the conventions of [31].”
A stochastic target formulation for optimal switching problems in finite horizon (2009, background): “Since the functions u, ũ and v need not be smooth, we shall appeal to the following classical notion of viscosity solution, see [6] for a general survey on this topic.”
GUARANTEED MINIMUM WITHDRAWAL BENEFIT IN VARIABLE ANNUITIES (2008, unknown): “Since the solution to equation (2.8) is expected to have a linear growth at infinity, the strong comparison principle holds in the sense of viscosity solution (Crandal et al. 1992; Barles et al.”
Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations (2007, background): “[13]), for every α > 0, there exist X, Y ∈ S(R) such that”
EXISTENCE AND COMPARISON RESULTS FOR FULLY NONLINEAR DEGENERATE ELLIPTIC EQUATIONS WITHOUT ZEROTH-ORDER TERM* (2001, methodology): “We refer the reader to the ‘‘Users’ guide’’ of Crandall, Ishii and Lions [10] for the introduction of these boundary conditions in the viscosity solutions’ sense and to Barles, Rouy and Souganidis [7] and Da Lio [11] for more specific discussions on the generalized Dirichlet problem for quasilinear ”
ON ∞-HARMONIC FUNCTIONS ON THE HEISENBERG GROUP (2002, methodology): “with the last inequality coming from the maximum principle in (3).”
On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations (2002, methodology): “We refer the reader to the articles of Lions [15–17] and the book by Fleming and Soner [8] for results in this direction and to the User’s guide [6] for a detailed presentation of the notion of viscosity solutions.”
A non-local regularization of first order Hamilton–Jacobi equations (2005, methodology): “Remark that P u(t, x) is the projection onR × R of the parabolic subjet ofu (see[12] for the definition of semi-jets).”
Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications (2005, unknown): “The proof of the comparison principle of viscosity solutions (see [13]) yields, for each α > 0, the estimate λj ṽj λj v̂λj + ρ(Lα−1) + L F,k,j in R, whereρ is as in (1.”
Function Spaces and Capacity Related to a Sublinear Expectation: Application to G-Brownian Motion Paths (2008, background): “We refer to [8] for the definition, existence, uniqueness and comparison theory of this type of parabolic PDE (see also [29] for our specific situation).”
Weak Solutions of Forward–Backward SDE's (2003, methodology): “For interested reader we refer to the standard references [8,16], and the recent paper [3] for some general discussions of the comparison principles for fully nonlinear PDEs.”
Oblique boundary value problems for augmented Hessian equations I (2015, background): “2) corresponds to that for viscosity solutions [4].”