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.

Let a continuous function satisfy both viscosity inequalities for the Laplace operator. On any relatively compact ball, take its harmonic replacement $h$ with matching boundary values. If $u-h$ has a positive maximum, the function $h+\epsilon (R^2-|x-a{|}^2)$ still lies below $u$ somewhere for sufficiently small positive $\epsilon$, while matching it on the boundary. At an interior maximum of their difference it is a valid upper test, but its Laplacian is $-2n\epsilon <0$, a contradiction. The opposite perturbation excludes a negative minimum. Thus $u=h$ throughout the ball and is a smooth harmonic function. A locally defined test is extended near its touching point using a smooth cutoff function when global tests are required.