Viscosity solution

Summary

A viscosity solution is a notion of weak solution for fully nonlinear first-order and second-order equations. Such equations, for example Hamilton–Jacobi equations, need not have classical solutions. It replaces derivatives of the solution, which may not exist, by derivatives of smooth test functions that touch the solution from above or from below. The notion was introduced by Crandall and Lions, and for many equations it gives existence, uniqueness and stability under uniform limits. Its name comes from the vanishing viscosity method. In this method a small multiple of the Laplacian is added to the equation and then sent to zero.