Calculus of Variations and Geometric Measure Theory

S. Di Marino - G. Speight

The p-Weak Gradient Depends on p

created by dimarino on 18 Nov 2013
modified on 24 Jul 2018


Published Paper

Inserted: 18 nov 2013
Last Updated: 24 jul 2018

Journal: Proceedings of the American Mathematical Society
Volume: 143
Number: 5239-5252
Year: 2014

ArXiv: 1311.4171 PDF

The new version generalizes the construction to $\mathbb{R}^{n}$ as suggested by a helpful referee. We also fix some minor typos, clarify terminology and update the references.


Given a>0, we construct a weighted Lebesgue measure on Rn for which the family of non constant curves has p-modulus zero for p\leq 1+a but the weight is a Muckenhoupt Ap weight for p>1+a. In particular, the p-weak gradient is trivial for small p but non trivial for large p. This answers an open question posed by several authors. We also give a full description of the p-weak gradient for any locally finite Borel measure on the real line.

Tags: GeMeThNES
Keywords: weighted Sobolev spaces, Lipschitz functions, Weak Gradient