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Fully anisotropic elliptic problems with minimally integrable data

  • Angela Alberico
  • Iwona Chlebicka
  • Andrea CianchiEmail author
  • Anna Zatorska-Goldstein
Article

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.

Mathematics Subject Classification

35J25 35J60 35B65 

Notes

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.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Angela Alberico
    • 3
  • Iwona Chlebicka
    • 2
  • Andrea Cianchi
    • 1
    • 3
    Email author
  • Anna Zatorska-Goldstein
    • 2
  1. 1.Dipartimento di Matematica e Informatica “U. Dini”Università di FirenzeFirenzeItaly
  2. 2.Institute of Applied Mathematics and MechanicsUniversity of WarsawWarsawPoland
  3. 3.Istituto per le Applicazioni del Calcolo “M. Picone”Consiglio Nazionale delle RicercheNaplesItaly

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