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Fully anisotropic elliptic problems with minimally integrable data

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Abstract

We investigate nonlinear elliptic Dirichlet problems whose growth is driven by a general anisotropic N-function, which is not necessarily of power-type and need not satisfy the \(\Delta _2\) nor the \(\nabla _2\)-condition. Fully anisotropic, non-reflexive Orlicz–Sobolev spaces provide a natural functional framework associated with these problems. Minimal integrability assumptions are detected on the datum on the right-hand side of the equation ensuring existence and uniqueness of weak solutions. When merely integrable, or even measure, data are allowed, existence of suitably further generalized solutions—in the approximable sense—is established. Their maximal regularity in Marcinkiewicz-type spaces is exhibited as well. Uniqueness of approximable solutions is also proved in case of \(L^1\)-data.

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Acknowledgements

We wish to thank the referee for his careful reading of the paper, for his valuable comments.

Funding

This research was partly funded by: (i) Italian Ministry of University and Research (MIUR), Research Project Prin 2015 “Partial differential equations and related analytic-geometric inequalities”, No. 2015HY8JCC; (ii) GNAMPA of the Italian INdAM—National Institute of High Mathematics (Grant No. not available); (iii) NCN Polish Grant No. 2016/23/D/ST1/01072.

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica “U. Dini”, Università di Firenze, Viale Morgagni 67/A, 50134, Firenze, Italy

    Andrea Cianchi

  2. Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097, Warsaw, Poland

    Iwona Chlebicka & Anna Zatorska-Goldstein

  3. Istituto per le Applicazioni del Calcolo “M. Picone”, Consiglio Nazionale delle Ricerche, Via Castellino 111, 80131, Naples, Italy

    Angela Alberico & Andrea Cianchi

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  1. Angela Alberico
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  2. Iwona Chlebicka
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  4. Anna Zatorska-Goldstein
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Correspondence to Andrea Cianchi.

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Communicated by N. Trudinger.

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Alberico, A., Chlebicka, I., Cianchi, A. et al. Fully anisotropic elliptic problems with minimally integrable data. Calc. Var. 58, 186 (2019). https://doi.org/10.1007/s00526-019-1627-8

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