Skip to main content
SpringerLink
Log in

Singular Poisson equations on Finsler–Hadamard manifolds

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

In the first part of the paper we study the reflexivity of Sobolev spaces on non-compact and not necessarily reversible Finsler manifolds. Then, by using direct methods in the calculus of variations, we establish uniqueness, location and rigidity results for singular Poisson equations involving the Finsler–Laplace operator on Finsler–Hadamard manifolds having finite reversibility constant.

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. Alvino, A., Ferone, V., Lions, P.-L., Trombetti, G.: Convex symmetrization and applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 14(2), 275–293 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ambrosio, L.; Colombo, M.; Di Marino, S.: Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope. arXiv:1212.3779

  3. Aubin, T.: Problèmes isopérimétriques et espaces de Sobolev. J. Diff. Geom. 11(4), 573–598 (1976)

    MathSciNet  MATH  Google Scholar 

  4. Baernstein II, A.: A unified approach to symmetrization. In: Alvino, A. et al.: (eds.) Partial Differential Equations of Elliptic Type. Symposia Matematica, vol. 35, pp. 47–91. Cambridge University Press, Cambridge (1995)

  5. Bao, D., Chern, S.S., Shen, Z.: Introduction to Riemann–Finsler geometry. In: Graduate Texts in Mathematics, vol. 200. Springer, New York (2000)

  6. Bao, D., Robles, C., Shen, Z.: Zermelo navigation on Riemannian manifolds. J. Diff. Geom. 66, 377–435 (2004)

    MathSciNet  MATH  Google Scholar 

  7. Bellettini, G., Paolini, M.: Anisotropic motion by mean curvature in the context of Finsler geometry. Hokkaido Math. J. 25(3), 537–566 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. Belloni, M., Ferone, V., Kawohl, B.: Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators. Z. Angew. Math. Phys. 54(5), 771–783 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Cianchi, A., Salani, P.: Overdetermined anisotropic elliptic problems. Math. Ann. 345(4), 859–881 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cobzaş, Ş.: Functional analysis in asymmetric normed spaces. In: Frontiers in Mathematics. Springer, Basel (2013)

  12. do Carmo, M.P.: Riemannian Geometry. Birkhäuser, Boston (1992)

    Book  MATH  Google Scholar 

  13. Egloff, D.: Uniform Finsler Hadamard manifolds. Ann. Inst. H. Poincaré Phys. Théor. 66(3), 323–357 (1997)

    MathSciNet  MATH  Google Scholar 

  14. Evans, L.C.: Partial differential equations, Graduate Studies in Mathematics, 2nd edn. AIMS (2010)

  15. Ferone, V., Kawohl, B.: Remarks on a Finsler–Laplacian. Proc. Am. Math. Soc. 137(1), 247–253 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ge, Y., Shen, Z.: Eigenvalues and eigenfunctions of metric measure manifolds. Proc. Lond. Math. Soc. (3) 82(3), 725–746 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hajlasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Amer. Math. Soc. 145, 1–101 (2000)

  18. Hebey, E.: Nonlinear analysis on manifolds: Sobolev spaces and inequalities. In: Courant Lecture Notes in Mathematics, vol. 5. New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence (1999)

  19. Kristály, A.; Rudas, I.J.: Sobolev spaces on the ball endowed with Funk-type metrics. Nonlinear Anal. doi:10.1016/j.na.2014.09.015 (2014)

  20. Rademacher, H.-B.: A sphere theorem for non-reversible Finsler metrics. Math. Ann. 328(3), 373–387 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Randers, G.: On an asymmetrical metric in the fourspace of general relativity. Phys. Rev. (2) 59, 195–199 (1941)

    Article  MathSciNet  Google Scholar 

  22. Ohta, S.: Vanishing \(S\)-curvature of Randers spaces. Differ. Geom. Appl. 29(2), 174–178 (2011)

    Article  MATH  Google Scholar 

  23. Ohta, S.: Uniform convexity and smoothness, and their applications in Finsler geometry. Math. Ann. 343(3), 669–699 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ohta, S., Sturm, K.-T.: Heat flow on Finsler manifolds. Commun. Pure Appl. Math. 62(10), 1386–1433 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Shen, Z.: Volume comparison and its applications in Riemann–Finsler geometry. Adv. Math. 128(2), 306–328 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  26. Shen, Z.: On Finsler geometry of submanifolds. Math. Ann. 311(3), 549–576 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. Shen, Z.: The non-linear Laplacian for Finsler manifolds. In: The theory of Finslerian Laplacians and applications, Mathematics and its Applications, vol. 459, pp. 187–198. Kluwer, Dordrecht (1998)

  28. Wang, G., Xia, C.: A characterization of the Wulff shape by an overdetermined anisotropic PDE. Arch. Ration. Mech. Anal. 199(1), 99–115 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wang, G., Xia, C.: A sharp lower bound for the first eigenvalue on Finsler manifolds. Ann. Inst. H. Poincaré Anal. Non Linéaire 30(6), 983–996 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wu, B.Y., Xin, Y.L.: Comparison theorems in Finsler geometry and their applications. Math. Ann. 337(1), 177–196 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Yuan, L.; Zhao, W.: Cheeger’s constant and the first eigenvalue of a closed Finsler manifold (2013). arXiv:1309.2115

  32. Zeidler, E.: Nonlinear functional analysis and its applications. III. Variational methods and optimization. Springer, New York (1985)

    Book  MATH  Google Scholar 

  33. Zhao, W., Shen, Y.: A universal volume comparison theorem for Finsler manifolds and related results. Can. J. Math. 65(6), 1401–1435 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

Research supported by a Grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project no. PN-II-ID-PCE-2011-3-0241. Cs. Farkas is also supported by the State of Hungary, co-financed by the European Social Fund in the framework of TÁMOP 4.2.4.A/2-11-1-2012-0001 ‘National Excellence Program’ and by Collegium Talentum. The research of A. Kristály is also supported by János Bolyai Research Scholarship.

Author information

Authors and Affiliations

  1. Faculty of Technical and Human Sciences, Sapientia Hungarian University of Transylvania, 547367, Tîrgu Mureş, Romania

    Csaba Farkas

  2. Department of Economics, Babeş-Bolyai University, 400593, Cluj-Napoca, Romania

    Alexandru Kristály

  3. Institute of Applied Mathematics, Óbuda University, Budapest, 1034, Hungary

    Alexandru Kristály

  4. Department of Mathematics, Babeş-Bolyai University, 400084, Cluj-Napoca, Romania

    Csaba Varga

Authors
  1. Csaba Farkas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexandru Kristály
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Csaba Varga
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexandru Kristály.

Additional information

Communicated by L. Ambrosio.

Dedicated to Professor Gheorghe Moroşanu on the occasion of his 65th birthday.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Farkas, C., Kristály, A. & Varga, C. Singular Poisson equations on Finsler–Hadamard manifolds. Calc. Var. 54, 1219–1241 (2015). https://doi.org/10.1007/s00526-015-0823-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-015-0823-4

Mathematics Subject Classification

Search

Navigation