Viscosity solution

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In mathematics, the viscosity solution concept was introduced in the early 1980's by Pierre-Louis Lions and Michael 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-Bellman 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

H(x,u,Du) = 0

over a domain x\in\Omega has a solution if we can find a function u(x) continuous and differentiable over the entire domain such that x, u and Du (the differential of u) satisfy the above equation at every point.

Under the viscosity solution concept, u need not be everywhere differentiable. There may be points where Du does not exist, i.e. there could be a kink in u and yet u satisfies the equation in an appropriate sense. Although Du may not exist at some point, the superdifferential D + u and the subdifferential D − u, to be defined below, are used in its place.

Definition 1. D^+ u(x_0) = \left\{ p: \limsup_{x_1\rightarrow x_0} \frac{u(x_1)-u(x_0)-p (x_1-x_0)}{|x_1-x_0|}\le 0 \right\}

Definition 2. D^- u(x_0) = \left\{ p: \liminf_{x_1\rightarrow x_0} \frac{u(x_1)-u(x_0)-p (x_1-x_0)}{|x_1-x_0|}\ge 0 \right\}

Definition 3. A continuous function u is a viscosity supersolution of the above PDE if

H(x,u(x),p)\le 0, \forall x \in \Omega, \forall p \in D^+ u(x)

Definition 4. A continuous function u is a viscosity subsolution of the above PDE if

H(x,u(x),p)\ge 0, \forall x \in \Omega, \forall p \in D^- u(x).

Definition 5. A continuous function u is a viscosity solution of the PDE if it is both a viscosity supersolution and a viscosity subsolution.

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