 Viscosity solution

For a partial differential equation $F(x,u,Du,D^{2}u)=0$ whose operator is nonincreasing in the Hessian variable, a continuous viscosity subsolution satisfies $F(x_0,u(x_0),D\varphi (x_0),D^2\varphi (x_0))\le 0$ whenever a smooth test function $\varphi$ touches it from above. A viscosity supersolution satisfies the reverse inequality for tests touching from below. A viscosity solution satisfies both conditions. For $F=-\mathrm{tr}{D}^{2}u$, these conditions are respectively $\Delta \varphi \ge 0$ and $\Delta \varphi \le 0$. They make sense even when $u$ has no classical second derivatives.

A viscosity solution is a type of weak solution to certain types of nonlinear partial differential equations (PDEs), particularly those of the Hamilton-Jacobi type. The concept is particularly useful in cases where classical solutions may not exist, such as when solutions may be discontinuous or exhibit other singular behaviors. ### Definition A viscosity solution satisfies the PDE in a "viscosity" sense, which means it adheres to a specific geometric interpretation involving test functions.