Poincar\'e and Sobolev inequalities with variable exponents and log-Holder continuity only at the boundary

TL;DR

This paper establishes Sobolev-Poincaré and Poincaré inequalities in variable Lebesgue spaces on bounded John domains under a new boundary log-Hölder condition on the exponent, allowing for discontinuities and weaker regularity assumptions.

Contribution

It introduces a boundary log-Hölder continuity condition for variable exponents, enabling new Sobolev and Poincaré inequalities with minimal regularity requirements.

Findings

Proves inequalities under weaker boundary regularity assumptions.

Shows the boundary log-Hölder condition is necessary for main results.

Provides an application to a degenerate p(·)-Laplacian Neumann problem.

Abstract

We prove Sobolev-Poincar\'e and Poincar\'e inequalities in variable Lebesgue spaces $L^{p(\cdot)}(\Omega)$, with $\Omega\subset{\mathbb R}^n$ a bounded John domain, with weaker regularity assumptions on the exponent $p(\cdot)$ that have been used previously. In particular, we require $p(\cdot)$ to satisfy a new \emph{boundary $\log$-H\"older condition} that imposes some logarithmic decay on the oscillation of $p(\cdot)$ towards the boundary of the domain. Some control over the interior oscillation of $p(\cdot)$ is also needed, but it is given by a very general condition that allows $p(\cdot)$ to be discontinuous at every point of $\Omega$. Our results follows from a local-to-global argument based on the continuity of certain Hardy type operators. We provide examples that show that our boundary $\log$-H\"older condition is essentially necessary for our main results. The same examples are adapted to show that this condition is not sufficient for other related inequalities. Finally, we give an application to a Neumann problem for a degenerate $p(\cdot)$-Laplacian.