Calculus of Variations and Geometric Measure Theory

L. Ambrosio - S. Di Marino - G. Savaré

On the duality between p-Modulus and probability measures

created by ambrosio on 06 Nov 2013
modified by dimarino on 24 Jul 2018


Submitted Paper

Inserted: 6 nov 2013
Last Updated: 24 jul 2018

Year: 2013

ArXiv: 1311.1381 PDF


Motivated by recent developments on calculus in metric measure spaces $(X,\mathsf d,\mathfrak m)$, we prove a general duality principle between Fuglede's notion of $p$-modulus for families of finite Borel measures in $(X,\mathsf d)$ and probability measures with barycenter in $L^q(X,\mathfrak m)$, with $q$ dual exponent of $p\in (1,\infty)$. We apply this general duality principle to study null sets for families of parametric and non-parametric curves in $X$. In the final part of the paper we provide a new proof, independent of optimal transportation, of the equivalence of notions of weak upper gradient based on $p$-Modulus (Koskela-MacManus '98, Shanmugalingam '00) and suitable probability measures in the space of curves (Ambrosio-Gigli-Savare '11)

Tags: GeMeThNES