Motivated by a “curvewise” approach, this project seeks to develop new tools and a finer understanding of certain structures and issues concerning Sobolev Spaces. Indeed, in a doubling space with a Poincaré inequality the theory is very well understood. However in a setting which is just complete and separable, much can be done. Previously the tools have been test plans, abstract differential structures and Gradient flows on Hilbert space, which give remarkable results. However, that approach is abstract, and does not yield directly insights in terms of modulus. For example, the differential constructed there is an abstract one.
With Elefterios we were able to give a general statement of representing the differential using curves and plans. This lead to a surprising differential structure and new tools which we will seek to apply in understanding further geometric questions.
This project seeks to build these concepts and new consequences just using basic Modulus techniques. Remarkably, old techniques can be improved and it seems much can be recovered with such hands on techniques. The advantage of this is complete and transparent proofs, and also an ability to reach the borderline exponent of p=1. It also raises a host of new and intriguing questions which could not have been tackled before.
- Sylvester Eriksson-Bique and Elefterios Soultanis, Curvewise characterizations of minimal upper gradients and the construction of a Sobolev differential, (https://arxiv.org/abs/2102.08097),2021.
- Sylvester Eriksson-Bique, Density of Lipschitz Functions in Energy, (arXiv:2012.01892) 2020