A $$\texttt {p}(\cdot )$$-Poincaré-type inequality for variable exponent Sobolev spaces with zero boundary values in Carnot groups

Abstract

We prove a \(\texttt {p}(\cdot )\)-Poincare-type inequality for variable exponent Sobolev spaces with zero boundary values in Carnot groups. We then establish the existence and uniqueness (up to a set of zero \(\texttt {p}(\cdot )\)-capacity) of a minimizer to the Dirichlet energy integral for the variable exponent case.