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 dynamic programming (the Hamilton–Jacobi–Bellman equation), differential games (the Hamilton–Jacobi–Isaacs equation) or front evolution problems,^[1][2] 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

${\displaystyle F(x,u,Du,D^{2}u)=0}$

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.

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 does not need to be everywhere differentiable. There may be points where either ${\displaystyle Du}$ or ${\displaystyle D^{2}u}$ does not exist and yet u satisfies the equation in an appropriate generalized 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.