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.

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^[3] or the definition using semi-jets in the Users Guide.^[4]

Degenerate elliptic

An equation ${\displaystyle F(x,u,Du,D^{2}u)=0}$ in a domain ${\displaystyle \Omega }$ is defined to be degenerate elliptic if for any two symmetric matrices ${\displaystyle X}$ and ${\displaystyle Y}$ such that ${\displaystyle Y-X}$ is positive definite, and any values of ${\displaystyle x\in \Omega }$, ${\displaystyle u\in \mathbb {R} }$ and ${\displaystyle p\in \mathbb {R} ^{n}}$, we have the inequality ${\displaystyle F(x,u,p,X)\geq F(x,u,p,Y)}$. For example, ${\displaystyle -\Delta u=0}$ (where ${\displaystyle \Delta }$ denotes the Laplacian) is degenerate elliptic since in this case, ${\displaystyle F(x,u,p,X)=-{\text{trace}}(X)}$, and the trace of ${\displaystyle X}$ is the sum of its eigenvalues. Any real first- order equation is degenerate elliptic.

Viscosity subsolution

An upper semicontinuous function ${\displaystyle u}$ in ${\displaystyle \Omega }$ is defined to be a subsolution of the above degenerate elliptic equation in the viscosity sense if for any point ${\displaystyle x_{0}\in \Omega }$ and any ${\displaystyle C^{2}}$ function ${\displaystyle \phi }$ such that ${\displaystyle \phi (x_{0})=u(x_{0})}$ and ${\displaystyle \phi \geq u}$ in a neighborhood of ${\displaystyle x_{0}}$, we have ${\displaystyle F(x_{0},\phi (x_{0}),D\phi (x_{0}),D^{2}\phi (x_{0}))\leq 0}$.

Viscosity supersolution

A lower semicontinuous function ${\displaystyle u}$ in ${\displaystyle \Omega }$ is defined to be a supersolution of the above degenerate elliptic equation in the viscosity sense if for any point ${\displaystyle x_{0}\in \Omega }$ and any ${\displaystyle C^{2}}$ function ${\displaystyle \phi }$ such that ${\displaystyle \phi (x_{0})=u(x_{0})}$ and ${\displaystyle \phi \leq u}$ in a neighborhood of ${\displaystyle x_{0}}$, we have ${\displaystyle F(x_{0},\phi (x_{0}),D\phi (x_{0}),D^{2}\phi (x_{0}))\geq 0}$.

Viscosity solution

A continuous function u is a viscosity solution of the PDE ${\displaystyle F(x,u,Du,D^{2}u)=0}$ in ${\displaystyle \Omega }$ 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 ${\displaystyle |u'(x)|=1}$, or ${\displaystyle F(u')=|u'|-1=0}$, on ${\displaystyle (-1,1)}$ with boundary conditions ${\displaystyle u(-1)=u(1)=0}$. Then, the function ${\displaystyle u(x)=1-|x|}$ is a viscosity solution.

Indeed, note that the boundary conditions are satisfied classically, and $$Zdroj:https://en.wikipedia.org?pojem=Viscosity_solutions
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