= Perron subfunction for the Poisson equation
{c}
{title2=$\Delta v\geq f$}
= Perron subfunctions for the Poisson equation
{c}
{synonym}
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 $\Delta v\geq f$. A superfunction reverses the comparison and has $\Delta v\leq f$ when $C^2$. Adding a boundary upper bound gives the family whose supremum defines the <Perron method for the Dirichlet problem>.
Back to article page
