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Removability of zero modular capacity sets

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Abstract

We introduce a notion of modular with a corresponding modular function space in order to build a modular capacity theory. We give two different definitions of capacity, one of them of variational type, the other one through either the modular of the test functions, or the modular of their gradients. We study, in both cases, the removability of sets of zero capacity in fairly general abstract Sobolev spaces with zero boundary values. As a key tool, we establish a modular Poincaré inequality. With the notion of modular function space in hands, we find a way to introduce a Banach function space, which allows to compare the zero capacity sets with respect to both notions. Thanks to this comparison, we characterize the compact sets of zero variational type capacity as removable sets. The paper is enriched with several examples, extending and unifying many results already known in literature in the settings of Musielak–Orlicz–Sobolev spaces, Lorentz–Sobolev spaces, variable exponent Sobolev spaces.

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Acknowledgements

The authors thank the referees for their careful reading of the paper. In the present version a gap discovered in the first revision of the paper has been corrected.

Funding

The research of the second author was partly funded by (i) FFABR 2017 - Fondo di finanziamento alle attività di base della ricerca (ii) GNAMPA: Progetto di ricerca 2019: “Stime a priori per il problema dell’ostacolo sotto ipotesi minimali di regolarità”.

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Authors and Affiliations

  1. Dipartimento di Architettura, Università di Napoli, Via Monteoliveto, 3, 80134, Napoli, Italy

    Alberto Fiorenza

  2. Istituto per le Applicazioni del Calcolo “Mauro Picone”, sezione di Napoli, Consiglio Nazionale delle Ricerche, via Pietro Castellino, 111, 80131, Napoli, Italy

    Alberto Fiorenza

  3. Dipartimento di Matematica ed Applicazioni “R. Caccioppoli”, Università di Napoli, Via Cintia, 80126, Napoli, Italy

    Flavia Giannetti

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  1. Alberto Fiorenza
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  2. Flavia Giannetti
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Correspondence to Flavia Giannetti.

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Fiorenza, A., Giannetti, F. Removability of zero modular capacity sets. Rev Mat Complut 34, 511–540 (2021). https://doi.org/10.1007/s13163-020-00361-z

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