Superharmonic function

Summary

A superharmonic function is a lower semicontinuous function whose value at each point is at least its average over spheres around that point. A function $u$ is superharmonic exactly when $-u$ is subharmonic, so the two theories are mirror images. The potentials of positive charges, such as the Newtonian kernel $\abs{x}^{2-n}$ in $\R^n$ for $n \ge 3$, are the basic examples. Superharmonic functions form the upper class of the Perron method and are the main objects of classical potential theory.