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, 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 has a solution if we can find a function u(x) continuous and differentiable over the entire domain such that , , , 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 or 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.
Definition
There are several equivalent ways to phrase the definition of viscosity solutions. See for example the section II.4 of Fleming and Soner's book or the definition using semi-jets in the Users Guide.
Degenerate elliptic An equation in a domain is defined to be degenerate elliptic if for any two symmetric matrices and such that is positive definite, and any values of , and , we have the inequality . For example, (where denotes the Laplacian) is degenerate elliptic since in this case, , and the trace of is the sum of its eigenvalues. Any real first- order equation is degenerate elliptic.
Viscosity subsolution An upper semicontinuous function in is defined to be a subsolution of the above degenerate elliptic equation in the viscosity sense if for any point and any function such that and in a neighborhood of , we have .
Viscosity supersolution A lower semicontinuous function in is defined to be a supersolution of the above degenerate elliptic equation in the viscosity sense if for any point and any function such that and in a neighborhood of , we have .
Viscosity solution A continuous function u is a viscosity solution of the PDE in if it is both a supersolution and a subsolution. Note that the boundary condition in the viscosity sense has not been discussed here.
Example
Consider the boundary value problem , or , on with boundary conditions . Then, the function is a viscosity solution.
Indeed, note that the boundary conditions are satisfied classically, and is well-defined in the interior except at . Thus, it remains to show that the conditions for viscosity subsolution and viscosity supersolution hold at . Suppose that is any function differentiable at with and near . From these assumptions, it follows that . For positive , this inequality implies , using that for . On the other hand, for , we have that . Because is differentiable, the left and right limits agree and are equal to , and we therefore conclude that , i.e., . Thus, is a viscosity subsolution. Moreover, the fact that is a supersolution holds vacuously, since there is no function differentiable at with and near . This implies that is a viscosity solution.
In fact, one may prove that is the unique viscosity solution for such problem. The uniqueness part involves a more refined argument.
Discussion
The previous boundary value problem is an eikonal equation in a single spatial dimension with , where the solution is known to be the signed distance function to the boundary of the domain. Note also in the previous example, the importance of the sign of . In particular, the viscosity solution to the PDE with the same boundary conditions is . This can be explained by observing that the solution is the limiting solution of the vanishing viscosity problem as goes to zero, while is the limit solution of the vanishing viscosity problem . One can readily confirm that solves the PDE for each . Further, the family of solutions converges toward the solution as vanishes (see Figure).