page_title: Viscosity solution
section_title: Discussion
hierarchical_section_title: Viscosity solution / Example / Discussion
context_page_description: 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. 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, differential games 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, 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.
context_section_description: 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 epsilon. Further, the family of solutions  converge toward the solution  as  vanishes (see Figure).
caption: Family of solutions 
    
      
        
          
            u
            
              ϵ
            
          
        
      
      {\displaystyle u_{\epsilon }}
    
   converging toward 
    
      
        
          u
          (
          x
          )
          =
          1
          −
          
            |
          
          x
          
            |
          
        
      
      {\displaystyle u(x)=1-|x|}
    
  .