 Perron subfunction for the Poisson equation

A continuous function $v$ is a Perron subfunction for $\Delta u=f$ when it lies below every classical solution of that Poisson equation on a relatively compact open ball whenever it lies below that solution on the ball's boundary. For $C^2$ functions, this is equivalent to $\Delta v\ge f$. A superfunction reverses the comparison and has $\Delta v\le f$ when $C^2$. Adding a boundary upper bound gives the family whose supremum defines the Perron method for the Dirichlet problem.