Global higher integrability and Hardy inequalities for double-phase functionals under a capacity density condition
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908287075614720 |
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| author | Bäuerlein, Fabian Riccò, Samuele Schätzler, Leah |
| author_facet | Bäuerlein, Fabian Riccò, Samuele Schätzler, Leah |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_21580 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global higher integrability and Hardy inequalities for double-phase functionals under a capacity density condition Bäuerlein, Fabian Riccò, Samuele Schätzler, Leah Analysis of PDEs 31B15, 35J66, 35J70, 49N60 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. |
| title | Global higher integrability and Hardy inequalities for double-phase functionals under a capacity density condition |
| topic | Analysis of PDEs 31B15, 35J66, 35J70, 49N60 |
| url | https://arxiv.org/abs/2503.21580 |