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Regularity properties for quasiminimizers of a (p, q)-Dirichlet integral

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Abstract

Using a variational approach we study interior regularity for quasiminimizers of a (p, q)-Dirichlet integral, as well as regularity results up to the boundary, in the setting of a metric space equipped with a doubling measure and supporting a Poincaré inequality. For the interior regularity, we use De Giorgi type conditions to show that quasiminimizers are locally Hölder continuous and they satisfy Harnack inequality, the strong maximum principle and Liouville’s Theorem. Furthermore, we give a pointwise estimate near a boundary point, as well as a sufficient condition for Hölder continuity and a Wiener type regularity condition for continuity up to the boundary. Finally, we consider (p, q)-minimizers and we give an estimate for their oscillation at boundary points.

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References

  1. Baroni, P., Colombo, M., Mingione, G.: Harnack inequalities for double phase functionals. Nonlinear Anal. 121, 206–222 (2015)

    Article  MathSciNet  Google Scholar 

  2. Björn, J.: Boundary continuity for quasiminimizers on metric spaces. Illinois J. Math. 46, 383–403 (2002)

    Article  MathSciNet  Google Scholar 

  3. Björn, J.: Sharp exponents and a Wiener type condition for boundary regularity of quasiminimizers. Adv. Math. 301, 804–819 (2016)

    Article  MathSciNet  Google Scholar 

  4. Björn, A., Björn, J.: Nonlinear Potential Theory on Metric Spaces, EMS Tracts in Mathematics, vol. 17. European Mathematical Society. EMS, Zurich (2011)

    Book  Google Scholar 

  5. Björn, A., Björn, J., Shanmugalingam, N.: The Dirichlet problem for p-harmonic functions on metric spaces. J. Reine Angew. Math. 556, 173–203 (2003)

    MathSciNet  MATH  Google Scholar 

  6. Björn, J., MacManus, P., Shanmugalingam, N.: Fat sets and pointwise boundary estimates for \(p\)-harmonic functions in metric spaces. J. Anal. Math. 85, 339–369 (2001)

    Article  MathSciNet  Google Scholar 

  7. Bögelein, V., Duzaar, F., Marcellini, P.: Parabolic equations with \(p, q\)-growth. J. Math. Pures Appl. 100, 535–563 (2013)

    Article  MathSciNet  Google Scholar 

  8. Bögelein, V., Duzaar, F., Marcellini, P., Scheven, C.: A variational approach to doubly nonlinear equations, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29, 739–772 (2018)

    Article  MathSciNet  Google Scholar 

  9. Cheeger, J.: Differentiability of Lipschitz functions on metric spaces. Geom. Funct. Anal. 9, 428–517 (1999)

    Article  MathSciNet  Google Scholar 

  10. Colombo, M., Mingione, G.: Regularity for double phase variational problems. Arch. Rat. Mech. Anal. 215, 443–496 (2015)

    Article  MathSciNet  Google Scholar 

  11. Colombo, M., Mingione, G.: Bounded minimisers of double phase variational integrals. Arch. Rat. Mech. Anal. 218, 219–273 (2015)

    Article  MathSciNet  Google Scholar 

  12. Cupini, G., Marcellini, P., Mascolo, E.: Existence and regularity for elliptic equations under \(p, q\)-growth. Adv. Differ. Equ. 19, 693–724 (2014)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  14. De Filippis, C., Oh, J.: Regularity for multi-phase variational problems. J. Differ. Equ. 267, 1631–1670 (2019)

    Article  MathSciNet  Google Scholar 

  15. De Giorgi, E.: Sulla differenziabilitá e l’analiticitá delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 3, 25–43 (1957)

  16. DiBenedetto, E., Gianazza, U.: A Wiener-type condition for boundary continuity of quasi-minima of variational integrals. Manuscripta Math. 149, 339–346 (2016)

    Article  MathSciNet  Google Scholar 

  17. DiBenedetto, E., Trudinger, N.S.: Harnack inequalities for quasiminima of variational integrals. Ann. Inst. H. Poincaré Anal. Non Linéaire 1(4), 295–308 (1984)

  18. Giaquinta, M., Giusti, E.: On the regularity of the minima of variational integrals. Acta Math. 148, 31–46 (1982)

    Article  MathSciNet  Google Scholar 

  19. Giaquinta, M., Giusti, E., Quasi-minima, Ann, Inst, H.: Poincar é Anal. Non Linéaire 1, 79–107 (1984)

  20. Giusti, E.: Direct Methods in the Calculus of Variations. World Scientific Publishing, River Edge (2003)

    Book  Google Scholar 

  21. Hadzhy, O.V., Skrypnik, I.I., Voitovych, M.V.: Interior continuity, continuity up to the boundary and Harnack’s inequality for double-phase elliptic equations with non-logarithmic conditions. arXiv:2012.10960v1

  22. Heinonen, J., Kilpelainen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, Oxford (1993)

    MATH  Google Scholar 

  23. Heinonen, J., Koskela, P., Shanmugalingam, N., Tyson, J.: Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper Gradients. New Mathematical Monographs. Cambridge University Press, 27 (2015)

  24. Keith, S., Zhong, X.: The Poincaré inequality is an open ended condition. Ann. Math. 167, 575–599 (2008)

    Article  MathSciNet  Google Scholar 

  25. Kinnunen, J., Shanmugalingam, N.: Regularity of quasi-minimizers on metric spaces. Manuscripta Math. 105, 401–423 (2001)

    Article  MathSciNet  Google Scholar 

  26. Krylov, N.V., Safonov, M.V.: Certain properties of solutions of parabolic equations with measurable coefficients, (Russian) Izv. Akad. Nauk SSSR 40, 161–175 (1980)

    Google Scholar 

  27. Marcellini, P.: Anisotropic and \(p\), \(q\)-nonlinear partial differential equations. Rend. Lincei Sci. Fis. Nat. 31, 295–301 (2020)

    Article  Google Scholar 

  28. Marcellini, P.: A variational approach to parabolic equations under general and \(p, q\)-growth conditions. Nonlinear Anal. 17, 194 (2020)

    MathSciNet  MATH  Google Scholar 

  29. Marcellini, P.: Growth conditions and regularity for weak solutions to nonlinear elliptic pdes. J. Math. Anal. Appl. (2021). https://doi.org/10.1016/j.jmaa.2020.124408

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  31. Mazya, V.G.: Sobolev Spaces. Springer, Berlin (1985)

    Google Scholar 

  32. Mingione, G., Rǎdulescu, V.: Recent developments in problems with nonstandard growth and nonuniform ellipticity. J. Math. Anal. Appl. (2021). https://doi.org/10.1016/j.jmaa.2021.125197

    Article  MathSciNet  MATH  Google Scholar 

  33. Nastasi, A.: Neumann p-Laplacian problems with a reaction term on metric spaces. Ric. Mat. (2020). https://doi.org/10.1007/s11587-020-00532-6

    Article  Google Scholar 

  34. Shanmugalingam, N.: Harmonic functions on metric spaces, Illinois J. Math

  35. Shanmugalingam, N.: Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoamericana 16(2), 243–279 (2000)

    Article  MathSciNet  Google Scholar 

  36. Tachikawa, A.: Boundary regularity of minimizers of double phase functionals. J. Math. Anal. Appl. (2021). https://doi.org/10.1016/j.jmaa.2020.123946

    Article  MathSciNet  MATH  Google Scholar 

  37. Ziemer, W.P.: Boundary regularity for quasiminima. Arch. Ration. Mech. Anal. 92, 371–382 (1986)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors wish to thank Professor Juha Kinnunen for supporting us in our research and for all the enlightening discussions and advice. Special thanks go to Professor Paolo Marcellini for the panoramical view he gently provided on the existing literature and open questions on the subject. The authors are grateful to the referees for their careful reading and the useful comments. The second author was supported by a doctoral training Grant for 2021 from the Väisälä Fund.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Palermo, Via Archirafi 34, 90123, Palermo, Italy

    Antonella Nastasi

  2. Department of Mathematics and Systems Analysis, Aalto University, Espoo, Finland

    Cintia Pacchiano Camacho

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  1. Antonella Nastasi
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  2. Cintia Pacchiano Camacho
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Correspondence to Antonella Nastasi.

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Communicated by Y. Giga.

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Nastasi, A., Pacchiano Camacho, C. Regularity properties for quasiminimizers of a (p, q)-Dirichlet integral. Calc. Var. 60, 227 (2021). https://doi.org/10.1007/s00526-021-02099-y

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