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, log in via an institution to check access.
Similar content being viewed by others
References
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)
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)
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)
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)
Björn, A., Björn, J.: Nonlinear Potential Theory on Metric Spaces, EMS Tracts in Mathematics, vol. 17. European Mathematical Society (EMS), Zürich (2011)
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)
Björn, A., Björn, J.: The variational capacity with respect to nonopen sets in metric spaces. Potential Anal. 40(1), 57–80 (2014)
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
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)
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)
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics Series. CRC Press, Boca Raton (1992)
Federer, H.: Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, vol. 153. Springer, New York (1969)
Giusti, E.: Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, vol. 80. Birkhäuser, Basel (1984)
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)
Hakkarainen, H., Kinnunen, J.: The BV-capacity in metric spaces. Manuscr. Math. 132(1–2), 51–73 (2010)
Hakkarainen, H., Korte, R., Lahti, P., Shanmugalingam, N.: Stability and continuity of functions of least gradient. Anal. Geom. Metr. Spaces 3, Art. 9 (2015)
Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations, Unabridged republication of the 1993 original. Dover Publications Inc, Mineola (2006)
Heinonen, J., Koskela, P.: Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181(1), 1–61 (1998)
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)
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)
Lahti, P.: A notion of fine continuity for BV functions on metric spaces. Potential Anal. 46(2), 279–294 (2017)
Lahti, P.: Strong approximation of sets of finite perimeter in metric spaces. Manuscr. Math. 1–20. doi:10.1007/s00229-017-0948-1
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)
Malý, J., Ziemer, W.: Fine Regularity of Solutions of Elliptic Partial Differential Equations, Mathematical Surveys and Monographs, 51. American Mathematical Society, Providence (1997)
Miranda Jr., M.: Functions of bounded variation on “good” metric spaces. J. Math. Pures Appl. (9) 82(8), 975–1004 (2003)
Shanmugalingam, N.: Harmonic functions on metric spaces. Ill. J. Math. 45(3), 1021–1050 (2001)
Shanmugalingam, N.: Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoam. 16(2), 243–279 (2000)
Ziemer, W.P.: Weakly differentiable functions. Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics, vol. 120. Springer, New York (1989)
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
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. Ambrosio.
Rights and permissions
About this article
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00526-017-1242-5