Properties of monotone mappings (Q1112318)

A map u, defined on a set \(D(u)\subseteq {\mathbb{R}}^ d\) with values in \({\mathbb{R}}^ d\) is called monotone, if \[ <(x-y),(u(x)-u(y))>\geq 0,\quad \forall x,y\in D(u). \] In the article, the usual questions of function theory are studied, as convergence, measurability, integrability, differentiability, transformation of size of monotone operators. In particular, it is shown, that monotone operators are locally bounded and measurable, and in the case they are additionally of bounded variation, they are almost everywhere differentiable. A theorem of Aleksandrov is also generalized.