Calculus of Variations and Geometric Measure Theory

E. Pasqualetto

Testing the Sobolev property with a single test plan

created by pasqualetto on 05 Jun 2020
modified on 21 Aug 2021


Accepted Paper

Inserted: 5 jun 2020
Last Updated: 21 aug 2021

Journal: Studia Mathematica
Pages: 23
Year: 2020


We prove that in a vast class of metric measure spaces (namely, those whose associated Sobolev space is separable) the following property holds: a single test plan can be used to recover the minimal weak upper gradient of any Sobolev function. This means that, in order to identify which are the exceptional curves in the weak upper gradient inequality, it suffices to consider the negligible sets of a suitable Borel measure on curves, rather than the ones of the $p$-modulus. Moreover, on $\sf RCD$ spaces we can improve our result, showing that the test plan can be also chosen to be concentrated on an equi-Lipschitz family of curves.

Keywords: Sobolev space, test plan, RCD space