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 Ω ⊂ Rn. 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 Ω, we establish an integral Hardy inequality. Further, we show that fatness of Rn \ Ω is equivalent to a boundary Poincar´e inequality, a pointwise Hardy inequality and to the local uniform p-fatness of Rn \ Ω. 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.