Global higher integrability and Hardy inequalities for double-phase functionals under a capacity density condition

Abstract: We prove global higher integrability for functionals of double-phase type under a uniform local capacity density condition on the complement of the considered domain $\Omega \subset \mathbb{R}^n$. In this context, we investigate a new natural notion of variational capacity associated to the double-phase integrand. Under the related fatness condition for the complement of $\Omega$, we establish an integral Hardy inequality. Further, we show that fatness of $\mathbb{R}^n \setminus \Omega$ is equivalent to a boundary Poincar\'e inequality, a pointwise Hardy inequality and to the local uniform $p$-fatness of $\mathbb{R}^n \setminus \Omega$. We provide a counterexample that shows that the expected Maz'ya type inequality - a key intermediate step toward global higher integrability - does not hold with the notion of capacity involving the double-phase functional itself.