= Viscosity solution
{wiki}
For a <partial differential equation> $F(x,u,Du,D^2u)=0$ whose operator is nonincreasing in the Hessian variable, a continuous viscosity subsolution satisfies $F(x_0,u(x_0),D\phi(x_0),D^2\phi(x_0))\leq0$ whenever a smooth <test function> $\phi$ touches it from above. A viscosity supersolution satisfies the reverse inequality for tests touching from below. A viscosity solution satisfies both conditions. For $F=-\operatorname{tr}D^2u$, these conditions are respectively $\Delta\phi\geq0$ and $\Delta\phi\leq0$. They make sense even when $u$ has no classical second derivatives.
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