Poincaré inequalities and Neumann problems for the variable exponent setting

David Cruz-Uribe; Michael Penrod; Scott Rodney; David Cruz-Uribe; Michael Penrod; Scott Rodney

doi:10.3934/mine.2022036

Mathematics in Engineering

2022, Volume 4, Issue 5: 1-22. doi: 10.3934/mine.2022036

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Research article

Poincaré inequalities and Neumann problems for the variable exponent setting

1.
Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA
2.
Department of Mathematics, Physics and Geology, Cape Breton University, Sydney, NS B1M1A2, CA

Received: 10 April 2021 Accepted: 17 August 2021 Published: 24 September 2021

In an earlier paper, Cruz-Uribe, Rodney and Rosta proved an equivalence between weighted Poincaré inequalities and the existence of weak solutions to a family of Neumann problems related to a degenerate $ p $-Laplacian. Here we prove a similar equivalence between Poincaré inequalities in variable exponent spaces and solutions to a degenerate $ {p(\cdot)} $-Laplacian, a non-linear elliptic equation with nonstandard growth conditions.
- degenerate Sobolev spaces,
- $ p $-Laplacian,
- Poincaréinequalities,
- variable exponent,
- nonstandard growth conditions
Citation: David Cruz-Uribe, Michael Penrod, Scott Rodney. Poincaré inequalities and Neumann problems for the variable exponent setting[J]. Mathematics in Engineering, 2022, 4(5): 1-22. doi: 10.3934/mine.2022036

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Abstract

In an earlier paper, Cruz-Uribe, Rodney and Rosta proved an equivalence between weighted Poincaré inequalities and the existence of weak solutions to a family of Neumann problems related to a degenerate $ p $-Laplacian. Here we prove a similar equivalence between Poincaré inequalities in variable exponent spaces and solutions to a degenerate $ {p(\cdot)} $-Laplacian, a non-linear elliptic equation with nonstandard growth conditions.

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