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.
has a solution if we can find a function u(x) continuous and differentiable over the entire domain such that
If a scalar equation is degenerate elliptic (defined below), one can define a type of weak solution called viscosity solution.
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.
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[3] or the definition using semi-jets in the Users Guide.
Indeed, note that the boundary conditions are satisfied classically, and
is differentiable, the left and right limits agree and are equal to
The uniqueness part involves a more refined argument.
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.
is the limiting solution of the vanishing viscosity problem
is the limit solution of the vanishing viscosity problem
The three basic properties of viscosity solutions are existence, uniqueness and stability.
The term viscosity solutions first appear in the work of Michael G. Crandall and Pierre-Louis Lions in 1983 regarding the Hamilton–Jacobi equation.
[6] The name is justified by the fact that the existence of solutions was obtained by the vanishing viscosity method.
The definition of solution had actually been given earlier by Lawrence C. Evans in 1980.
[9] Subsequently the definition and properties of viscosity solutions for the Hamilton–Jacobi equation were refined in a joint work by Crandall, Evans and Lions in 1984.
[10] For a few years the work on viscosity solutions concentrated on first order equations because it was not known whether second order elliptic equations would have a unique viscosity solution except in very particular cases.
The breakthrough result came with the method introduced by Robert Jensen in 1988 to prove the comparison principle using a regularized approximation of the solution which has a second derivative almost everywhere (in modern versions of the proof this is achieved with sup-convolutions and Alexandrov theorem).
[11] In subsequent years the concept of viscosity solution has become increasingly prevalent in analysis of degenerate elliptic PDE.
Based on their stability properties, Barles and Souganidis obtained a very simple and general proof of convergence of finite difference schemes.
[12] Further regularity properties of viscosity solutions were obtained, especially in the uniformly elliptic case with the work of Luis Caffarelli.
[13] Viscosity solutions have become a central concept in the study of elliptic PDE.
In particular, Viscosity solutions are essential in the study of the infinity Laplacian.
[14] In the modern approach, the existence of solutions is obtained most often through the Perron method.
[4] The vanishing viscosity method is not practical for second order equations in general since the addition of artificial viscosity does not guarantee the existence of a classical solution.
Other names that were suggested were Crandall–Lions solutions, in honor to their pioneers,
