In mathematics, the viscosity solution concept was introduced in the early 1980s by Pierre-Louis Lions and Michael G. Crandall as a generalization of the classical concept of what is meant by a 'solution' to a partial differential equation (PDE). It has been found that the viscosity solution is the natural solution concept to use in many applications of PDE's, including for example first order equations arising in optimal control (the Hamilton-Jacobi equation), differential games (the Isaacs equation) or front evolution problems, as well as second-order equations such as the ones arising in stochastic optimal control or stochastic differential games.
The classical concept was that a PDE
over a domain has a solution if we can find a function u(x) continuous and differentiable over the entire domain such that , , , satisfy the above equation at every point.
If a scalar equation is degenerate elliptic (defined below), one can define a type of weak solution called viscosity solution. Under the viscosity solution concept, u need not be everywhere differentiable. There may be points where either or does not exist and yet u satisfies the equation in an appropriate sense. The definition allows only for certain kind of singularities, so that existence, uniqueness, and stability under uniform limits, hold for a large class of equations.
There are several equivalent ways to phrase the definition of viscosity solutions. See for example the section II.4 of Fleming and Soner's book or the definition using semi-jets in the Users Guide.
An equation in a domain is defined to be degenerate elliptic if for any two symmetric matrices and such that is positive definite, and any values of , and , we have the inequality . For example is degenerate elliptic. Any first order equation is degenerate elliptic.
An upper semicontinuous function in is defined to be a subsolution of a degenerate elliptic equation in the viscosity sense if for any point and any function such that and in a neighborhood of , we have .
A lower semicontinuous function in is defined to be a supersolution of a degenerate elliptic equation in the viscosity sense if for any point and any function such that and in a neighborhood of , we have .
A continuous function u is a viscosity solution of the PDE if it is both a viscosity supersolution and a viscosity subsolution.
The three basic properties of viscosity solutions are existence, uniqueness and stability.
- The uniqueness of solutions requires some extra structural assumptions on the equation. Yet it can be shown for a very large class of degenerate elliptic equations. It is a direct consequence of the comparison principle. Some simple examples where comparison principle holds are
- with H uniformly continuous in x.
- (Uniformly elliptic case) so that is Lipschitz with respect to all variableas and for every and , for some .
- The existence of solutions holds in all cases where the comparison principle holds and the boundary conditions can be enforced in some way (through barrier functions in the case of a Dirichlet boundary condition). For first order equations, it can be obtained using the vanishing viscosity method  or for most equations using Perron's method.
- The stability of solutions in holds as follows: a locally uniform limit of a sequence of solutions (or subsolutions, or supersolutions) is a solution (or subsolution, or supersolution).
The term viscosity solutions first appear in the work of Michael G. Crandall and Pierre-Louis Lions in 1983  regarding the Hamilton-Jacobi equation. The name is justified by the fact that the existence of solutions was obtained by the vanishing viscosity method. The definition of solution had actually been given earlier by Lawrence Evans in 1980. Subsequently the definition and properties of viscosity solutions for the Hamilton-Jacobi equation were refined in a joint work by Crandall, Evans and Lions in 1984.
For a few years the work on viscosity solutions concentrated on first order equations because it was not known whether second order elliptic equations would have a unique viscosity solution except in very particular cases. The breakthrough result came with the method introduced by Robert Jensen in 1988  to prove the comparison principle using a regularized approximation of the solution which has a second derivative almost everywhere (in modern versions of the proof this is achieved with sup-convolutions and Alexandrov theorem).
In subsequent years the concept of viscosity solution has become increasingly prevalent in analysis of degenerate elliptic PDE. Based on their stability properties, Barles and Souganidis obtained a very simple and general proof of convergence of finite difference schemes. Further regularity properties of viscosity solutions were obtained, especially in the uniformly elliptic case with the work of Luis Caffarelli. Viscosity solutions have become a central concept in the study of elliptic PDE as can be corroborated by the fact that currently the Users guide  has more than 800 citations, being the most cited paper of mathematics for six years straight from 2003 to 2008 according to mathscinet.
In the modern approach, the existence of solutions is obtained most often though the Perron method. The vanishing viscosity method is not practical for second order equations in general since the addition of artificial viscosity does not guarantee the existence of a classical solution. Moreover, the definition of viscosity solutions does not involve any viscosity of any kind. Thus, it has been suggested that the name viscosity solution does not represent the concept appropriately. Yet, the name persists because of the history of the subject. Other names that were suggested were Crandall-Lions solutions, in honor to their pioneers, -weak solutions, referring to their stability properties, or comparison solutions, referring to their most characteristic property.
- I. Dolcetta and P. Lions, eds., (1995), Viscosity Solutions and Applications. Springer, ISBN 978-3-540-62910-8.
- Wendell H. Fleming, H. M . Soner., eds., (2006), Controlled Markov Processes and Viscosity Solutions. Springer, ISBN 978-0-387-26045-7.
- Crandall, Michael G.; Ishii, Hitoshi; Lions, Pierre-Louis (1992), "User's guide to viscosity solutions of second order partial differential equations", American Mathematical Society. Bulletin. New Series 27 (1): 1–67, doi:10.1090/S0273-0979-1992-00266-5, ISSN 0002-9904
- Crandall, Michael G.; Lions, Pierre-Louis (1983), "Viscosity solutions of Hamilton-Jacobi equations", Transactions of the American Mathematical Society 277 (1): 1–42, doi:10.2307/1999343, ISSN 0002-9947
- Ishii, Hitoshi (1987), "Perron's method for Hamilton-Jacobi equations", Duke Mathematical Journal 55 (2): 369–384, doi:10.1215/S0012-7094-87-05521-9, ISSN 0012-7094
- Ishii, Hitoshi (1989), "On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs", Communications on Pure and Applied Mathematics 42 (1): 15–45, doi:10.1002/cpa.3160420103, ISSN 0010-3640
- Evans, Lawrence C. (1980), "On solving certain nonlinear partial differential equations by accretive operator methods", Israel Journal of Mathematics 36 (3): 225–247, doi:10.1007/BF02762047, ISSN 0021-2172
- Crandall, Michael G.; Evans, Lawrence C.; Lions, Pierre-Louis (1984), "Some properties of viscosity solutions of Hamilton-Jacobi equations", Transactions of the American Mathematical Society 282 (2): 487–502, doi:10.2307/1999247, ISSN 0002-9947
- Jensen, Robert (1988), "The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations", Archive for Rational Mechanics and Analysis 101 (1): 1–27, doi:10.1007/BF00281780, ISSN 0003-9527
- Barles, G.; Souganidis, P. E. (1991), "Convergence of approximation schemes for fully nonlinear second order equations", Asymptotic Analysis 4 (3): 271–283, ISSN 0921-7134
- Caffarelli, Luis A.; Cabré, Xavier (1995), Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications 43, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0437-7