Home Poincaré inequality and energy of separating sets
Article
Licensed
Unlicensed Requires Authentication

Poincaré inequality and energy of separating sets

  • Emanuele Caputo and Nicola Cavallucci EMAIL logo
Published/Copyright: March 28, 2025

Abstract

We study geometric characterizations of the Poincaré inequality in doubling metric measure spaces in terms of properties of separating sets. Given a couple of points and a set separating them, such properties are formulated in terms of several possible notions of energy of the boundary, involving for instance the perimeter, codimension type Hausdorff measures, capacity, Minkowski content and approximate modulus of suitable families of curves. We prove the equivalence within each of these conditions and the 1-Poincaré inequality.

MSC 2020: 30L15; 53C23; 49J52

Communicated by Juha Kinnunen


Award Identifier / Grant number: 948021

Funding statement: The first author was supported by the Academy of Finland (Grant No. 321896). He currently acknowledges the support by the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 948021).

Acknowledgements

The first author also thanks the kind hospitality of the University of Ottawa and of Augusto Gerolin. We thank Francesco Nobili for a careful reading of the manuscript.

References

[1] L. Ambrosio, Fine properties of sets of finite perimeter in doubling metric measure spaces, Set-Valued. Anal. 10 (2002), no. 2, 111–128. 10.1023/A:1016548402502Search in Google Scholar

[2] L. Ambrosio and S. Di Marino, Equivalent definitions of BV space and of total variation on metric measure spaces, J. Funct. Anal. 266 (2014), no. 7, 4150–4188. 10.1016/j.jfa.2014.02.002Search in Google Scholar

[3] L. Ambrosio, S. Di Marino and N. Gigli, Perimeter as relaxed Minkowski content in metric measure spaces, Nonlinear Anal. 153 (2017), 78–88. 10.1016/j.na.2016.03.010Search in Google Scholar

[4] L. Ambrosio, M. Miranda, Jr. and D. Pallara, Special functions of bounded variation in doubling metric measure spaces, Calculus of Variations: Topics from the Mathematical Heritage of E. De Giorgi, Quad. Mat. 14, Seconda Università degli Studi di Napoli, Caserta (2004), 1–45. Search in Google Scholar

[5] A. Björn and J. Björn, Nonlinear Potential Theory on Metric Spaces, Vol. 17, European Mathematical Society, Zürich, 2011. 10.4171/099Search in Google Scholar

[6] P. Bonicatto, E. Pasqualetto and T. Rajala, Indecomposable sets of finite perimeter in doubling metric measure spaces, Calc. Var. Partial Differential Equations 59 (2020), no. 2, Paper No. 63. 10.1007/s00526-020-1725-7Search in Google Scholar

[7] C. Brena, F. Nobili and E. Pasqualetto, Maps of bounded variation from PI spaces to metric spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) (2023), 10.2422/2036-2145.202407_012. 10.2422/2036-2145.202407_012Search in Google Scholar

[8] E. Caputo, Geometric characterizations of PI spaces: An overview of some modern techniques, preprint (2025), https://www.arxiv.org/abs/2501.19132. Search in Google Scholar

[9] E. Caputo and N. Cavallucci, A geometric approach to Poincaré inequality and Minkowski content of separating sets, Int. Math. Res. Not. IMRN 2025 (2025), no. 1, 1–30. 10.1093/imrn/rnae276Search in Google Scholar

[10] E. Caputo, J. Koivu and T. Rajala, Sobolev, BV and perimeter extensions in metric measure spaces, Ann. Fenn. Math. 49 (2024), no. 1, 135–165. 10.54330/afm.143899Search in Google Scholar

[11] E. Caputo and T. Rossi, First-order heat content asymptotics on 𝖱𝖢𝖣 ( K , N ) spaces, Nonlinear Anal. 238 (2024), Article ID 113385. 10.1016/j.na.2023.113385Search in Google Scholar

[12] N. Cavallucci and A. Sambusetti, Packing and doubling in metric spaces with curvature bounded above, Math. Z. 300 (2022), no. 3, 3269–3314. 10.1007/s00209-021-02905-5Search in Google Scholar

[13] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), no. 3, 428–517. 10.1007/s000390050094Search in Google Scholar

[14] E. Durand-Cartagena, S. Eriksson-Bique, R. Korte and N. Shanmugalingam, Equivalence of two BV classes of functions in metric spaces, and existence of a Semmes family of curves under a 1-Poincaré inequality, Adv. Calc. Var. 14 (2021), no. 2, 231–245. 10.1515/acv-2018-0056Search in Google Scholar

[15] S. Eriksson-Bique, Characterizing spaces satisfying Poincaré inequalities and applications to differentiability, Geom. Funct. Anal. 29 (2019), no. 1, 119–189. 10.1007/s00039-019-00479-3Search in Google Scholar

[16] S. Eriksson-Bique and J. Gong, Almost uniform domains and Poincaré inequalities, Trans. London Math. Soc. 8 (2021), no. 1, 243–298. 10.1112/tlm3.12032Search in Google Scholar

[17] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Textb. Math., CRC Press, Boca Raton, 2015. 10.1201/b18333Search in Google Scholar

[18] K. Fässler and T. Orponen, Metric currents and the Poincaré inequality, Calc. Var. Partial Differential Equations 58 (2019), no. 2, Paper No. 69. 10.1007/s00526-019-1514-3Search in Google Scholar

[19] P. Hajłasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688, 1–101. 10.1090/memo/0688Search in Google Scholar

[20] J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext, Springer, New York, 2001. 10.1007/978-1-4613-0131-8Search in Google Scholar

[21] J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), no. 1, 1–61. 10.1007/BF02392747Search in Google Scholar

[22] J. Heinonen, P. Koskela, N. Shanmugalingam and J. T. Tyson, Sobolev Spaces on Metric Measure Spaces, New Math. Monogr. 27, Cambridge University, Cambridge, 2015. 10.1017/CBO9781316135914Search in Google Scholar

[23] V. Honzlová Exnerová, J. Malý and O. Martio, Modulus in Banach function spaces, Ark. Mat. 55 (2017), no. 1, 105–130. 10.4310/ARKIV.2017.v55.n1.a5Search in Google Scholar

[24] V. Honzlová-Exnerová, J. Malý and O. Martio, AM-modulus and Hausdorff measure of codimension one in metric measure spaces, Math. Nachr. 295 (2022), no. 1, 140–157. 10.1002/mana.202000059Search in Google Scholar

[25] S. Keith, Modulus and the Poincaré inequality on metric measure spaces, Math. Z. 245 (2003), no. 2, 255–292. 10.1007/s00209-003-0542-ySearch in Google Scholar

[26] R. Korte and P. Lahti, Relative isoperimetric inequalities and sufficient conditions for finite perimeter on metric spaces, Ann. Inst. H. Poincaré C Anal. Non Linéaire 31 (2014), no. 1, 129–154. 10.1016/j.anihpc.2013.01.005Search in Google Scholar

[27] R. Korte, P. Lahti, X. Li and N. Shanmugalingam, Notions of Dirichlet problem for functions of least gradient in metric measure spaces, Rev. Mat. Iberoam. 35 (2019), no. 6, 1603–1648. 10.4171/rmi/1095Search in Google Scholar

[28] P. Lahti, Federer’s characterization of sets of finite perimeter in metric spaces, Anal. PDE 13 (2020), no. 5, 1501–1519. 10.2140/apde.2020.13.1501Search in Google Scholar

[29] P. Lahti and N. Shanmugalingam, Fine properties and a notion of quasicontinuity for BV functions on metric spaces, J. Math. Pures Appl. (9) 107 (2017), no. 2, 150–182. 10.1016/j.matpur.2016.06.002Search in Google Scholar

[30] M. Miranda, Jr., Functions of bounded variation on “good” metric spaces, J. Math. Pures Appl. (9) 82 (2003), no. 8, 975–1004. 10.1016/S0021-7824(03)00036-9Search in Google Scholar

Received: 2024-10-09
Revised: 2025-02-11
Published Online: 2025-03-28
Published in Print: 2025-07-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 6.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/acv-2024-0109/html
Scroll to top button