Weakly Monotone Functions on Metric Measure Spaces

Abstract: In this paper we develop a theory of weakly monotone functions on metric measure spaces. We develop some analytic properties of such functions and prove an approximation theorem for a general $N^{1,p}(X)$ function for $p>1$. We exhibit a class of functionals whose minimizers are weakly monotone which includes the $p-$energy functional and as a corollary we show that minimizers of this energy satisfy a weak maximum and weak minimum principle.