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Neumann p-Laplacian problems with a reaction term on metric spaces

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Abstract

We use a variational approach to study existence and regularity of solutions for a Neumann p-Laplacian problem with a reaction term on metric spaces equipped with a doubling measure and supporting a Poincaré inequality. Trace theorems for functions with bounded variation are applied in the definition of the variational functional and minimizers are shown to satisfy De Giorgi type conditions.

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References

  1. Ambrosio, L.: Fine properties of sets of finite perimeter in doubling metric measure spaces, calculus of variations, nonsmooth analysis and related topics. Set-Valued Anal. 10(2–3), 111–128 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ambrosio, L., Miranda Jr., M., Pallara, D.: Special functions of bounded variation in doubling metric measure spaces. Calculus of variations: topics from the mathematical heritage of E. De Giorgi, pp. 1–45 (2004)

  3. Björn, A., Björn, J.: Nonlinear potential theory on metric spaces. EMS Tracts in Mathematics, 17, European Mathematical Society. EMS, Zurich (2011)

  4. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

  7. Durand-Cartagena, E., Lemenant, A.: Some stability results under domain variation for Neumann problems in metric spaces. Ann. Acad. Sci. Fenn. Math. 35(2), 537–563 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Franchi, B., Hajlasz, P., Koskela, P.: Definitions of Sobolev classes on metric spaces. Ann. Inst. Fourier (Grenoble) 49, 1903–1924 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  10. Hajlasz, P.: Sobolev spaces on metric-measure spaces. Contemp. Math. 338, 173–218 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hakkarainen, H., Kinnunen, J., Lahti, P., Lehtelä, P.: Relaxation and integral representation for functionals of linear growth on metric measure spaces. Anal. Geom. Metr. Spaces 4, 288–313 (2016)

    MathSciNet  MATH  Google Scholar 

  12. Heinonen, J., Koskela, P.: From local to global in quasiconformal structures. Proc. Natl. Acad. Sci. USA 93, 554–556 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. Heinonen, J., Koskela, P.: Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181, 1–61 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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)

  15. Kinnunen, J., Martio, O.: Nonlinear potential theory on metric spaces. Illinois J. Math. 46, 857–883 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Kinnunen, J., Korte, R., Shanmugalingam, N., Tuominen, H.: A characterization of Newtonian functions with zero boundary values. Calc. Var. Partial Differ. Equ. 43, 507–528 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lahti, P.: Extensions and traces of functions of bounded variation on metric spaces. J. Math. Anal. Appl. 423(1), 521–537 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lahti, P.: Federer’s characterization of sets of finite perimeter in metric spaces. arXiv:1906.03125

  20. Lahti, P., Li, X., Wang, Z.: Traces of Newton-Sobolev, Hajlasz-Sobolev, and BV functions on metric spaces. arXiv:1911.00533

  21. Lahti, P., Malý, L., Shanmugalingam, N.: An Analog of the Neumann Problem for the 1-Laplace Equation in the Metric Setting: Existence, Boundary Regularity, and Stability. arXiv:1708.02346

  22. Lahti, P., Shanmugalingam, N.: Trace theorems for functions of bounded variation in metric setting. J. Funct. Anal. 274(10), 2754–2791 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Malý, L.: Trace and extension theorems for Sobolev-type functions in metric spaces. arXiv:1704.06344

  24. Malý, L., Shanmugalingam, N.: Neumann problem for p -Laplace equation in metric spaces using a variational approach: existence, boundedness, and boundary regularity. J. Differ. Equ. 265, 2431–2460 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  25. Miranda Jr., M.: Functions of bounded variation on “good” metric spaces. J. Math. Pures Appl. 82(9), 975–1004 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. Nastasi, A.: Weak solution for Neumann \((p, q)\)-Laplacian problem on Riemannian manifold. J. Math. Anal. Appl. 479(1), 45–61 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Antonella Nastasi

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  1. Antonella Nastasi
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Nastasi, A. Neumann p-Laplacian problems with a reaction term on metric spaces. Ricerche mat 71, 415–430 (2022). https://doi.org/10.1007/s11587-020-00532-6

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  • DOI: https://doi.org/10.1007/s11587-020-00532-6

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