<abstract xmlns="http://www.w3.org/1999/xhtml"><p>In this article we define the weighted variable exponent-Sobolev spaces on arbitrary metric spaces, with finite diameter and equipped with finite, positive Borel regular outer measure. We employ a Hajlasz definition, which uses a point wise maximal inequality. We prove that these spaces are Banach, that the Poincaré inequality holds and that lipschitz functions are dense. We develop a capacity theory based on these spaces. We study basic properties of capacity and several convergence results. As an application, we prove that each weighted variable exponent-Sobolev function has a quasi-continuous representative, we study different definitions of the first order weighted variable exponent-Sobolev spaces with zero boundary values, we define the Dirichlet energy and we prove that it has a minimizer in the weighted variable exponent -Sobolev spaces case.</p></abstract>

In this article we define the weighted variable exponent-Sobolev spaces on arbitrary metric spaces, with finite diameter and equipped with finite, positive Borel regular outer measure. We employ a Hajlasz definition, which uses a point wise maximal inequality. We prove that these spaces are Banach, that the Poincaré inequality holds and that lipschitz functions are dense. We develop a capacity theory based on these spaces. We study basic properties of capacity and several convergence results. As an application, we prove that each weighted variable exponent-Sobolev function has a quasi-continuous representative, we study different definitions of the first order weighted variable exponent-Sobolev spaces with zero boundary values, we define the Dirichlet energy and we prove that it has a minimizer in the weighted variable exponent -Sobolev spaces case.

Weighted Variable Exponent Sobolev spaces on metric measure spaces

Moroccan Journal of Pure and Applied Analysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"Weighted Variable Exponent Sobolev spaces on metric measure spaces"}

In this article we define the weighted variable exponent-Sobolev spaces on arbitrary metric spaces, with finite diameter and equipped with finite, positive...

Weighted Variable Exponent Sobolev spaces on metric measure spaces

Published Online: May 16, 2019

Page range: 62 - 76

Received: Feb 06, 2019

Accepted: Apr 03, 2019

DOI: https://doi.org/10.1515/mjpaa-2018-0007

Keywords
weighted variable exponent Sobolev spaces on metric spaces, capacity, weighted variable exponent Sobolev with zero boundary values, Dirichlet energy

© 2018 Moulay Cherif Hassib et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Weighted Variable Exponent Sobolev spaces on metric measure spaces

Published Online: May 16, 2019

Page range: 62 - 76

Received: Feb 06, 2019

Accepted: Apr 03, 2019

DOI: https://doi.org/10.1515/mjpaa-2018-0007

Keywordsweighted variable exponent Sobolev spaces on metric spaces, capacity, weighted variable exponent Sobolev with zero boundary values, Dirichlet energy

© 2018 Moulay Cherif Hassib et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
weighted variable exponent Sobolev spaces on metric spaces, capacity, weighted variable exponent Sobolev with zero boundary values, Dirichlet energy