Viscosity solution

In mathematics, the viscosity solution concept was introduced in the early 1980s 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, [I. Dolcetta and P. Lions, eds., (1995), "Viscosity Solutions and Applications." Springer, ISBN 978-3-540-62910-8.] 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 $xinOmega$ 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. $D^+\; u(x\_0)\; =\; left\{\; p:\; limsup\_\{x\_1\; ightarrow\; x\_0\}\; frac\{u(x\_1)-u(x\_0)-p\; (x\_1-x\_0)\}le\; 0\; ight\}$

Definition 2. $D^-\; u(x\_0)\; =\; left\{\; p:\; liminf\_\{x\_1\; ightarrow\; x\_0\}\; frac\{u(x\_1)-u(x\_0)-p\; (x\_1-x\_0)\}ge\; 0\; ight\}$

Roughly speaking, every $,p,$ in the set $,D^+\; u,$ is an upper bound on the "slope" of $,u,$ at $,x\_0,$, and every $,p,$ in the set $,D^-\; u,$ is a lower bound on the "slope" of $,u,$ at $,x\_0,$.

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.

There is an equivalent definition for viscosity solution without using the definition of sub (super) differentials. See the detail at the section II.4 of Fleming and Soner's book. [Wendell H. Fleming, H. M . Soner., eds., (2006), "Controlled Markov Processes and Viscosity Solutions." Springer, ISBN 978-0387-260457.]

References