Liouville's theorem and the restricted mean value property in the plane
Let \(U\) be a domain in \(\mathbb{R}^2\) such that the complement to \(U\) is polar and let \(r\) be a real function on \(U\) such that \(0<r\leq\| \cdot\|+M\) where \(M\) is a nonnegative constant. A nonnegative numerical function \(f\) on \(U\) is called \(r\)-supermedian if, for every \(x\in U\), the average of \(f\) on the disk of centre \(x\) and radius \(r(x)\) is smaller than or equal to \(f(x)\). The current paper provides a short proof of the fact that every lower semicontinuous \(r\)-supermedian function is constant.
scientific article; zbMATH DE number 1248319