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Boundedness of solutions to Dirichlet, Neumann and Robin problems for elliptic equations in Orlicz spaces

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Abstract

Boundary value problems for second-order elliptic equations in divergence form, whose nonlinearity is governed by a convex function of non-necessarily power type, are considered. The global boundedness of their solutions is established under boundary conditions of Dirichlet, or Neumann, or Robin type. A decisive role in the results is played by optimal forms of Orlicz-Sobolev embeddings and boundary trace embeddings, which allow for critical growths of the coefficients.

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Acknowledgements

We wish to thank the referee for carefully reading the paper and for her/his valuable comments.

Funding

This research was partly funded by: (i) Research Project of the Italian Ministry of Education, University and Research (MIUR), Prin 2017 “Nonlinear differential problems via variational, topological and set-valued methods”, Grant Number 2017AYM8XW (G.Barletta). (ii) Research Project of the Italian Ministry of Education, University and Research (MIUR) Prin 2017 “Direct and inverse problems for partial differential equations: theoretical aspects and applications”, Grant Number 201758MTR2 (A.Cianchi); (iii) GNAMPA of the Italian INdAM - National Institute of High Mathematics (Grant Number not available) (G.Barletta, A.Cianchi, G.Marino); (iv) DFG via grant GZ: MA 10100/1-1, Grant Number 496629752 (G.Marino).

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Authors and Affiliations

  1. Dipartimento di Ingegneria Civile, dell’Energia, dell’Ambiente e dei Materiali, Università Mediterranea di Reggio Calabria, Via Zehender, 89122, Reggio Calabria, Italy

    Giuseppina Barletta

  2. Dipartimento di Matematica e Informatica U. Dini, Università di Firenze, Viale Morgagni 67/A, 50134, Florence, Italy

    Andrea Cianchi

  3. Institut für Mathematik, Universität Augsburg, Universitätsstrasse 2, 86159, Augsburg, Germany

    Greta Marino

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  1. Giuseppina Barletta
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  2. Andrea Cianchi
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Correspondence to Andrea Cianchi.

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Communicated by L. Szekelyhidi.

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Barletta, G., Cianchi, A. & Marino, G. Boundedness of solutions to Dirichlet, Neumann and Robin problems for elliptic equations in Orlicz spaces. Calc. Var. 62, 65 (2023). https://doi.org/10.1007/s00526-022-02393-3

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