##  [Viscosity Solution](/viscosity-solution-0) 

 Definition

A viscosity solution is a generalized notion of solution for certain nonlinear first- and second-order partial differential equations defined by comparison with smooth test functions rather than by pointwise classical derivatives.

 

 

 

 

 

 





## Principle

Principle

Characterize admissible sub- and supersolutions by local tangency with C^∞ test functions and enforce a comparison principle that selects a unique continuous solution under suitable conditions.

 

 

 

 

 





## Demonstration

Demonstration

For the Hamilton–Jacobi equation u_t + H(x,Du)=0 on R^n, a continuous function u is a viscosity subsolution if for every smooth φ such that u−φ has a local maximum at x0, we have φ_t(x0)+H(x0,Dφ(x0))≤0 (and analogously for supersolutions).

 

 

 

 

## Misapplication

Misapplication

Treating a viscosity solution as a distributional or Sobolev weak solution and applying integration-by-parts identities that require derivatives almost everywhere can lead to incorrect conclusions about uniqueness or regularity.

 

 

 

 

 





## Consequence

Consequence

When the comparison principle holds, existence and uniqueness follow for the viscosity solution framework; stability under uniform limits and monotone approximation schemes is obtained.

 

 

 

 

## Reversal

Reversal

A classical solution satisfies the PDE pointwise with required derivatives — if a classical solution exists, it is also a viscosity solution; the reversal highlights that viscosity solutions extend classical ones rather than replace them.

 

 

 

 

 





## Boundary

Boundary

Applies to fully nonlinear PDEs where classical solutions may not exist; it is not a substitute for distributional solution concepts in linear PDEs where different weak formulations are standard.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension exists with the notion of distributional (weak) solution: viscosity solutions rely on pointwise comparison with smooth tests, whereas distributional solutions use integral identities and are often tied to functional-analytic spaces.

 

 

 

 

 





## Synthesis

Synthesis

A viscosity solution is the comparison-based, stability-oriented generalized solution concept for certain nonlinear PDEs that extends classical solutions and ensures well-posedness when derivatives fail to exist.