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Quasilinear PDEs, Interpolation Spaces and Hölderian mappings

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Abstract

As in the work of Tartar [59], we develop here some new results on nonlinear interpolation of α-Hölderian mappings between normed spaces, by studying the action of the mappings on K-functionals and between interpolation spaces with logarithm functions. We apply these results to obtain some regularity results on the gradient of the solutions to quasilinear equations of the form

$$-\text{div}(\widehat{a}(\nabla u))+V(u)=f,$$

where V is a nonlinear potential and f belongs to non-standard spaces like Lorentz–Zygmund spaces. We show several results; for instance, that the mapping \(\cal{T}:\cal{T}f=\nabla u\) is locally or globally α-Hölderian under suitable values of α and appropriate hypotheses on V and â.

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Acknowledgements

A. Gogatishvili started work on this project during his visit to J. M. Rakotoson to the Laboratory of Mathematics of the University of Poitiers, France, in April 2022. He thanks for his generous hospitality and helpful atmosphere during the visit. He has been partially supported by the Czech Academy of Sciences (RVO 67985840), by the Czech Science Foundation (GAČR), grant no. 23-04720S, by the Shota Rustaveli National Science Foundation (SRNSF), grant no. FR-21-12353, and by the grant Minstry of Education and Science of the Republic of Kazakhstan (project no. AP14869887).

M.R. Formica is member of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and member of the UMI group “Teoria dell’Approssimazione e Applicazioni (T.A.A.)” and is partially supported by the INdAM-GNAMPA project, Risultati di regolarità per PDEs in spazi di funzione non-standard, codice CUP_E53C22001930001 and partially supported by University of Naples “Parthenope”, Dept. of Economic and Legal Studies, project CoRNDiS, DM MUR 737/2021, CUP I55F21003620001.

The authors thank the anonymous referee for the careful reading of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Sukkur IBA University, Sukkur, Pakistan

    I. Ahmed

  2. Università di Napoli “Federico II”, via Monteoliveto 3, 80134, Napoli, Italy

    A. Fiorenza

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

    A. Fiorenza

  4. Università degli Studi di Napoli “Parthenope”, via Generale Parisi 13, 80132, Napoli, Italy

    M. R. Formica

  5. Institute of Mathematics of the Czech Academy of Sciences, Zitná, 115 67, Prague 1, Czech Republic

    A. Gogatishvili

  6. Departement de Mathématiques et Laboratoire LaSIE, Université de La Rochelle, Av. Michel Crépeau, 17042, La Rochelle Cedex 1, France

    A. El Hamidi

  7. Laboratoire de Mathématiques et Applications, Université de Poitiers, 11 Bd Marie et Pierre Curie, Téléport 2, 86073, Poitiers Cedex 9, France

    J. M. Rakotoson

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  1. I. Ahmed
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Dedicated to Oleg V. Besov on the occasion of his 90th birthday

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Ahmed, I., Fiorenza, A., Formica, M.R. et al. Quasilinear PDEs, Interpolation Spaces and Hölderian mappings. Anal Math 49, 895–950 (2023). https://doi.org/10.1007/s10476-023-0245-z

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