Global higher integrability and Hardy inequalities for double-phase functionals under a capacity density condition
Fabian Bäuerlein, Samuele Riccò, Leah Schätzler
TL;DR
The paper develops a capacity-density framework for variational problems with double-phase growth, introducing infimal $\varphi$-capacity and related fatness notions to capture how the complement of a domain influences global regularity. It proves an integral Hardy inequality under fatness, establishes the equivalence of fatness with boundary Poincaré and pointwise Hardy inequalities, and demonstrates self-improving properties of infimal $\varphi$-fatness. Using a Maz'ya-type and Gehring-type arguments, it derives global higher integrability results for quasi-minimizers of functionals with growth $\varphi(x,t)=t^p+a(x)t^q$, under suitable $(p,q)$-growth conditions and domain fatness. A counterexample shows that a Maz'ya-type inequality may fail when using the capacity linked directly to the double-phase, motivating the infimal-capacity approach and highlighting the delicate interaction between domain geometry and the two-branch growth.
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 $Ω\subset \mathbb{R}^n$. 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 $Ω$, we establish an integral Hardy inequality. Further, we show that fatness of $\mathbb{R}^n \setminus Ω$ is equivalent to a boundary Poincaré inequality, a pointwise Hardy inequality and to the local uniform $p$-fatness of $\mathbb{R}^n \setminus Ω$. 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.