Skip to main content
Log in

An optimal transportation problem related to the limits of solutions of local and nonlocal \(p\)-Laplace-type problems

  • Published:
Revista Matemática Complutense Aims and scope Submit manuscript

Abstract

We investigate the asymptotic behavior of minimizers of problems related to the \(p\)-Laplace equation \(-\Delta _p u=f\). As \(p\rightarrow \infty \), the minimizers converge (up to a subsequence) to a function, which maximizes the functional \(I(u)= \int fu\) with appropriate constraints. The main result of this paper is that the problem of maximizing \(I\) with Dirichlet boundary condition can be identified as a dual problem of a certain mass transportation problem. Our approach applies to the limits of both \(p\)-Laplace type problems in classical Sobolev spaces and analogous nonlocal problems in fractional Sobolev spaces.

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

Access this article

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. Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9(3), 428–517 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (1998)

  4. Evans, L.C., Gangbo, W.: Differential equations methods for the Monge–Kantorovich mass transfer problem. Mem. Am. Math. Soc. 137(653) (1999)

  5. García-Azorero, J., Manfredi, J.J., Peral, I., Rossi, J.D.: The Neumann problem for the \(\infty \)-Laplacian and the Monge–Kantorovich mass transfer problem. Bol. Soc. Esp. Mat. Apl. S\(\vec{e}\)MA 43, 7–28 (2008)

  6. Ishii, H., Loreti, P.: Limits of solutions of \(p\)-Laplace equations as \(p\) goes to infinity and related variational problems. SIAM J. Math. Anal. 37(2), 411–437 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Loss, M., Sloane, C.: Hardy inequalities for fractional integrals on general domains. J. Funct. Anal. 259(6), 1369–1379 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  8. Mazón, J.M., Rossi, J.D., Toledo, J.: An optimal transportation problem with a cost given by the Euclidean distance plus import/export taxes on the boundary. Revista Matemática Iberoamericana. Preprint. http://cvgmt.sns.it/paper/1876/

  9. Talenti, G.: Inequalities in rearrangement invariant function spaces. Nonlinear Anal. Funct. Spaces Appl. 5, 177–230 (Prague, 1994)

    Google Scholar 

  10. Villani, C.: Optimal Transportation, Old and New. Grundlehren der mathematische Wissenschaften. Springer, Berlin (2009)

    Google Scholar 

  11. Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence (2003)

Download references

Acknowledgments

The author would like to thank Petri Juutinen for introducing the author to the problems discussed in this paper and Julio Rossi for sharing his ideas about how to generalize the results. The author would also like to thank the reviewers for their helpful comments. The author has been supported by University of Jyväskylä and Väisälä Foundation.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Heikki Jylhä.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jylhä, H. An optimal transportation problem related to the limits of solutions of local and nonlocal \(p\)-Laplace-type problems. Rev Mat Complut 28, 85–121 (2015). https://doi.org/10.1007/s13163-014-0147-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13163-014-0147-5

Keywords

Mathematics Subject Classification (1991)

Navigation