Viscosity solution

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 ${\displaystyle x\in \Omega }$ has a solution if we can find a function u(x) continuous and differentiable over the entire domain such that ${\displaystyle x}$, ${\displaystyle u}$, ${\displaystyle Du}$, ${\displaystyle D^{2}u}$ satisfy the above equation at every point.

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