Skip to main content

A Federer-style characterization of sets of finite perimeter on metric spaces

Abstract

In the setting of a metric space equipped with a doubling measure that supports a Poincaré inequality, we show that a set E is of finite perimeter if and only if \({\mathcal {H}}(\partial ^1 I_E)<\infty \), that is, if and only if the codimension one Hausdorff measure of the 1-fine boundary of the set’s measure theoretic interior \(I_E\) is finite. To obtain the necessity of the above condition, we prove a suitable characterization of the 1-fine boundary, analogously to what is known in the case \(p>1\), and apply a quasicontinuity-type result for \(\mathrm {BV}\) functions proved in the metric setting by Lahti and Shanmugalingam (J Math Pures Appl (9) 107(2):150–182, 2017). To obtain the sufficiency, we generalize further results of fine potential theory from the case \(p>1\) to the case \(p=1\), including weak analogs of a Cartan property for solutions of obstacle problems, and of the Choquet property for finely open sets.

This is a preview of subscription content, access via your institution.

References

  1. 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)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000)

    MATH  Google Scholar 

  3. 3.

    Ambrosio, L., Miranda, M. Jr., Pallara, D.: Special functions of bounded variation in doubling metric measure spaces. In: Calculus of variations: topics from the mathematical heritage of E. De Giorgi, 1–45, Quad. Mat., 14, Dept. Math., Seconda Univ. Napoli, Caserta (2004)

  4. 4.

    Ambrosio, L., Tilli, P.: Topics on Analysis in Metric Spaces, Oxford Lecture Series in Mathematics and its Applications, vol. 25. Oxford University Press, Oxford (2004)

    MATH  Google Scholar 

  5. 5.

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

    Book  MATH  Google Scholar 

  6. 6.

    Björn, A., Björn, J.: Obstacle and Dirichlet problems on arbitrary nonopen sets in metric spaces, and fine topology. Rev. Mat. Iberoam. 31(1), 161–214 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Björn, A., Björn, J.: The variational capacity with respect to nonopen sets in metric spaces. Potential Anal. 40(1), 57–80 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Björn, A., Björn, J., Latvala, V.: The Cartan, Choquet and Kellogg properties for the fine topology on metric spaces. J. Anal. Math. arXiv:1410.5167

  9. 9.

    Björn, A., Björn, J., Latvala, V.: The weak Cartan property for the p-fine topology on metric spaces. Indiana Univ. Math. J. 64(3), 915–941 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Carriero, M., Dal Maso, G., Leaci, A., Pascali, E.: Relaxation of the nonparametric plateau problem with an obstacle. J. Math. Pures Appl. (9) 67(4), 359–396 (1988)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics Series. CRC Press, Boca Raton (1992)

    MATH  Google Scholar 

  12. 12.

    Federer, H.: Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, vol. 153. Springer, New York (1969)

    Google Scholar 

  13. 13.

    Giusti, E.: Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, vol. 80. Birkhäuser, Basel (1984)

    Book  MATH  Google Scholar 

  14. 14.

    Hajłasz, P.: Sobolev Spaces on Metric-Measure Spaces, Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces (Paris, 2002): Contemporary Mathematics, vol. 338, pp. 173–218. American Mathematical Society, Providence (2003)

    MATH  Google Scholar 

  15. 15.

    Hakkarainen, H., Kinnunen, J.: The BV-capacity in metric spaces. Manuscr. Math. 132(1–2), 51–73 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Hakkarainen, H., Korte, R., Lahti, P., Shanmugalingam, N.: Stability and continuity of functions of least gradient. Anal. Geom. Metr. Spaces 3, Art. 9 (2015)

  17. 17.

    Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations, Unabridged republication of the 1993 original. Dover Publications Inc, Mineola (2006)

    MATH  Google Scholar 

  18. 18.

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

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

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

    Book  MATH  Google Scholar 

  20. 20.

    Korte, R., Lahti, P.: Relative isoperimetric inequalities and sufficient conditions for finite perimeter on metric spaces. Ann. Inst. H. Poincaré Anal. Non Linéaire 31(1), 129–154 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Lahti, P.: A notion of fine continuity for BV functions on metric spaces. Potential Anal. 46(2), 279–294 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Lahti, P.: Strong approximation of sets of finite perimeter in metric spaces. Manuscr. Math. 1–20. doi:10.1007/s00229-017-0948-1

  23. 23.

    Lahti, P., Shanmugalingam, N.: Fine properties and a notion of quasicontinuity for BV functions on metric spaces. J. Math. Pures Appl. (9) 107(2), 150–182 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Malý, J., Ziemer, W.: Fine Regularity of Solutions of Elliptic Partial Differential Equations, Mathematical Surveys and Monographs, 51. American Mathematical Society, Providence (1997)

    Book  MATH  Google Scholar 

  25. 25.

    Miranda Jr., M.: Functions of bounded variation on “good” metric spaces. J. Math. Pures Appl. (9) 82(8), 975–1004 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Shanmugalingam, N.: Harmonic functions on metric spaces. Ill. J. Math. 45(3), 1021–1050 (2001)

    MathSciNet  MATH  Google Scholar 

  27. 27.

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

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Ziemer, W.P.: Weakly differentiable functions. Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics, vol. 120. Springer, New York (1989)

    MATH  Google Scholar 

Download references

Acknowledgements

The research was funded by a grant from the Finnish Cultural Foundation. The author wishes to thank Nageswari Shanmugalingam for reading the manuscript and providing useful comments.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Panu Lahti.

Additional information

Communicated by  L. Ambrosio.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lahti, P. A Federer-style characterization of sets of finite perimeter on metric spaces. Calc. Var. 56, 150 (2017). https://doi.org/10.1007/s00526-017-1242-5

Download citation

Mathematics Subject Classification

  • 30L99
  • 31E05
  • 26B30