History
The term viscosity solutions first appear in the work of Michael 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.
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