Skip to main content
SpringerLink
Log in

Local Boundedness for Minimizers of Anisotropic Functionals with Monomial Weights

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We study the local boundedness of minimizers of non uniformly elliptic integral functionals with a suitable anisotropic \(p,q-\) growth condition. More precisely, the growth condition of the integrand function \(f(x,\nabla u)\) from below involves different \(p_i>1\) powers of the partial derivatives of u and some monomial weights \(|x_i|^{\alpha _i p_i}\) with \(\alpha _i \in [0,1)\) that may degenerate to zero. Otherwise from above it is controlled by a q power of the modulus of the gradient of u with \(q\ge \max _i p_i\) and an unbounded weight \(\mu (x)\). The main tool in the proof is an anisotropic Sobolev inequality with respect to the weights \(|x_i|^{\alpha _i p_i}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams, R.A.: Sobolev Spaces. In: Pure and Applied Mathematics, vol. 65. Academic Press, New York (1975)

    Google Scholar 

  2. Bella, P., Schäffner, M.: Local boundedness and Harnack inequality for solutions of linear non-uniformly elliptic equations. Comm. Pure App. Math. 74(3), 451–675 (2021)

    Article  Google Scholar 

  3. Boccardo, L., Marcellini, P., Sbordone, C.: \(L^\infty \)-regularity for variational problems with sharp non standard growth conditions. Bollettino U.M.I. 7(4), 219–226 (1990)

    Google Scholar 

  4. Brandolini B., Cîrstea F.C.: Boundedness of solutions to singular anisotropic elliptic equations. Discrete Contin. Dyn. Syst. Ser. S 17(4), 1545–1561(2024)

  5. Cabré, X., Ros-Oton, X.: Sobolev and isoperimetric inequalities with monomial weights. J. Differ. Eq. 255, 4312–4336 (2013)

    Article  MathSciNet  Google Scholar 

  6. Cianchi, A.: Symmetrization in anisotropic elliptic problems. Comm. Partial Differ. Eq. 32, 693–717 (2007)

    Article  MathSciNet  Google Scholar 

  7. Cianchi, A.: Local boundedness of minimizers of anisotropic functionals. Ann. Inst. H. Poincaré C Anal. Non Linéaire 17, 147–168 (2000)

    Article  MathSciNet  Google Scholar 

  8. Cupini, G., Marcellini, P., Mascolo, E.: Regularity under sharp anisotropic general growth conditions. Discrete Contin. Dyn. Syst. Ser. B 11, 66–86 (2009)

    MathSciNet  Google Scholar 

  9. Cupini, G., Marcellini, P., Mascolo, E.: Regularity of minimizers under limit growth conditions. Nonlinear Anal. 153, 294–310 (2017)

    Article  MathSciNet  Google Scholar 

  10. Cupini, G., Marcellini, P., Mascolo, E.: Nonuniformly elliptic energy integrals with \(p, q\)-growth, Nonlinear. Analysis 177, 312–324 (2018)

    MathSciNet  Google Scholar 

  11. Cupini, G., Marcellini, P., Mascolo, E.: Local boundedness of weak solutions to elliptic equations with \(p, q\)-growth. Math. Eng. 5(3), 1–28 (2023)

  12. Cupini, G., Marcellini, P., Mascolo, E., Passarelli di Napoli, A.: Lipschitz regularity for degenerate elliptic integrals with \(p, q\)-growth. Adv. Calc. Var. 16(2), 443–465 (2023)

    Article  MathSciNet  Google Scholar 

  13. De Rosa, M.P., Grimaldi, A.G.: A Local boundedness result for a class of obstacle problems with non-standard growth conditions. J. Optim. Theory Appl. 195, 282–296 (2022)

    Article  MathSciNet  Google Scholar 

  14. di Blasio, G., Feo, F., Zecca, G.: Regularity results for local solutions to some anisotropic elliptic equations. Isr. J, Math (2023)

    Book  Google Scholar 

  15. Fabes, E., Kenig, C., Serapioni, R.: The local regularity of solutions of degenerate elliptic equations. Comm. Partial Differ. Eq. 7, 77–116 (1982)

    Article  MathSciNet  Google Scholar 

  16. Feo, F., Martin, J., Posteraro, M.R.: Sobolev anisotropic inequalities with monomial weights. J. Math. Anal. Appl. 505, 125557 (2022)

    Article  MathSciNet  Google Scholar 

  17. Fusco, N., Sbordone, C.: Some remarks on the regularity of minima of anisotropic integrals. Commun. Partial Differ. Eq. 18, 153–167 (1993)

    Article  MathSciNet  Google Scholar 

  18. Giusti, E.: Direct methods in the calculus of variations. World Scientific Publishing Co., Inc., River Edge, NJ, viii+403 pp. (2003)

  19. Hirsch, J., Schäffner, M.: Growth conditions and regularity, an optimal local boundedness result. Commun. Contemp. Math. 23(3), 2050029 (2021)

    Article  MathSciNet  Google Scholar 

  20. Marcellini, P.: Un example de solution discontinue d’un probleme variationnel dans le cas scalaire. Preprint Universitá di Firenze, (1987)

  21. Marcellini, P.: Regularity and existence of solutions of elliptic equations with \(p-q\)-growth conditions. J. Differ. Eq. 90, 1–30 (1991)

    Article  MathSciNet  Google Scholar 

  22. Marcellini, P.: Regularity for elliptic equations with general growth conditions. J. Differ. Eq. 105, 296–333 (1993)

    Article  MathSciNet  Google Scholar 

  23. Marcellini, P.: Regularity for some scalar variational problems under general growth conditions. J. Optim. Theory Appl. 90, 161–181 (1996)

    Article  MathSciNet  Google Scholar 

  24. Mingione, G., Rădulescu, V.: Recent developments in problems with nonstandard growth and nonuniform ellipticity. J. Math. Anal. Appl. 501, 125197 (2021)

    Article  MathSciNet  Google Scholar 

  25. Tartar, L.: Imbedding theorems of Sobolev spaces into Lorentz spaces. BUMI Serie 8, 479–500 (1998)

    MathSciNet  Google Scholar 

  26. Troisi, M.: Teoremi di inclusione per spazi di Sobolev non isotropi. Ricerche Mat. 18, 3–24 (1969)

Download references

Acknowledgements

The authors are partially supported by GNAMPA of the Italian INdAM (National Institute of High Mathematics) through the INdAM-GNAMPA Project Project 23 “Su alcuni problemi di regolarità del Calcolo delle Variazioni con convessità degenere” and “Equazioni di diffusione e analisi geometrica” CUP\(\_\)E53C22001930001 and INdAM-GNAMPA Project 24 “Disuguaglianze analitiche e geometriche” and “Interazione ottimale tra la regolarità dei coefficienti e l’anisotropia del problema in funzionali" CUP\(\_\)E53C23001670001. A. Passarelli di Napoli has been also partially supported by the Universitá degli Studi di Napoli “Federico II” through the project FRA-000022-ALTRI-CDA-752021-FRA-PASSARELLI and by the PNNR project Centro Nazionale per la Mobilitá Sostenibile (CNMS) code CN00000023, CUP E63C22000930007 (Spoke 10 Logistica Merci). F. Feo and M. R: Posteraro have been also partially supported by "Geometric-Analytic Methods for PDEs and Applications" (GAMPA) project CUP I53D23002420006 -funded by European Union - Next Generation EU within the PRIN 2022 program (D.D. 104 - 02/02/2022 Ministero dell’Università e della Ricerca). M. R. Posteraro has been also partially supported by "Direct and inverse problems for partial differential equations: theoretical aspects and applications" project funded by Ministero dell’Università e della Ricerca within the PRIN 2017 program. This manuscript reflects only the authors’ views and opinions and the Ministry cannot be considered responsible for them.

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria, Università degli Studi di Napoli “Parthenope”, Centro Direzionale Isola C4, 80143, Napoli, Italy

    Filomena Feo

  2. Dipartimento di Matematica e Appl. “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Complesso Monte S. Angelo, Via Cintia, Napoli, Italy

    Antonia Passarelli di Napoli & Maria Rosaria Posteraro

Authors
  1. Filomena Feo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Antonia Passarelli di Napoli
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Maria Rosaria Posteraro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maria Rosaria Posteraro.

Additional information

Communicated by Sofia Giuffre.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Feo, F., Passarelli di Napoli, A. & Posteraro, M.R. Local Boundedness for Minimizers of Anisotropic Functionals with Monomial Weights. J Optim Theory Appl (2024). https://doi.org/10.1007/s10957-024-02432-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10957-024-02432-3

Keywords

Mathematics Subject Classification

Search

Navigation