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