Relative isoperimetric inequalities and sufficient conditions for finite perimeter on metric spaces
Korte, Riikka ; Lahti, Panu
Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014), p. 129-154 / Harvested from Numdam

Nous étudions l'équivalence entre l'inégalité de Poincaré et plusieurs différentes inégalités isopérimétriques relatives sur les espaces métriques mesurés. Nous utilisons ensuite ces inégalités afin d'établir des conditions suffisantes sur le périmètre fini d'ensembles.

We study equivalence between the Poincaré inequality and several different relative isoperimetric inequalities on metric measure spaces. We then use these inequalities to establish sufficient conditions for the finite perimeter of sets.

Publié le : 2014-01-01
DOI : https://doi.org/10.1016/j.anihpc.2013.01.005
Classification:  28A12,  26A45,  30L99
@article{AIHPC_2014__31_1_129_0,
     author = {Korte, Riikka and Lahti, Panu},
     title = {Relative isoperimetric inequalities and sufficient conditions for finite perimeter on metric spaces},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {31},
     year = {2014},
     pages = {129-154},
     doi = {10.1016/j.anihpc.2013.01.005},
     mrnumber = {3165282},
     zbl = {1285.28003},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2014__31_1_129_0}
}
Korte, Riikka; Lahti, Panu. Relative isoperimetric inequalities and sufficient conditions for finite perimeter on metric spaces. Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) pp. 129-154. doi : 10.1016/j.anihpc.2013.01.005. http://gdmltest.u-ga.fr/item/AIHPC_2014__31_1_129_0/

[1] D. Aalto, J. Kinnunen, The discrete maximal operator in metric spaces, J. Anal. Math. 111 (2010), 369-390 | MR 2747071 | Zbl 1210.42029

[2] L. Ambrosio, Fine properties of sets of finite perimeter in doubling metric measure spaces, Calculus of Variations, Nonsmooth Analysis and Related Topics Set-Valued Anal. 10 no. 2–3 (2002), 111-128 | MR 1926376 | Zbl 1037.28002

[3] L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., The Clarendon Press, Oxford University Press, New York (2000) | MR 1857292 | Zbl 0957.49001

[4] L. Ambrosio, M. Miranda, 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. vol. 14, Dept. Math., Seconda Univ. Napoli, Caserta (2004), 1-45 | MR 2118414 | Zbl 1089.49039

[5] L. Ambrosio, P. Tilli, Topics on Analysis in Metric Spaces, Oxford Lecture Ser. Math. Appl. vol. 25, Oxford University Press, Oxford (2004) | MR 2039660 | Zbl 1080.28001

[6] A. Björn, J. Björn, Nonlinear Potential Theory on Metric Spaces, EMS Tracts in Mathematics vol. 17 (2011) | MR 2867756 | Zbl 1231.31001

[7] A. Björn, J. Björn, N. Shanmugalingam, Quasicontinuity of Newton–Sobolev functions and density of Lipschitz functions on metric spaces, Houston J. Math. 34 no. 4 (2008), 1197-1211 | MR 2465375 | Zbl 1170.46032

[8] S.G. Bobkov, C. Houdré, Some connections between isoperimetric and Sobolev-type inequalities, Mem. Amer. Math. Soc. 129 no. 616 (1997) | MR 1396954 | Zbl 0894.60021

[9] S. Buckley, Is the maximal function of a Lipschitz function continuous?, Ann. Acad. Sci. Fenn. Math. 24 no. 2 (1999), 519-528 | MR 1724375

[10] C.S. Camfield, Comparison of BV norms in weighted Euclidean spaces and metric measure spaces, PhD thesis, University of Cincinnati, 2008. | MR 2712069

[11] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 no. 3 (1999), 428-517 | MR 1708448 | Zbl 0942.58018

[12] M. Csörnyei, J. Grahl, T. O'Neil, Points of middle density in the real line, Real Anal. Exchange 37 no. 2 (2012), 243-248 | MR 3080589 | Zbl 1276.28009

[13] L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, Stud. Adv. Math., CRC Press, Boca Raton, FL (1992) | MR 1158660 | Zbl 0626.49007

[14] H. Federer, Geometric Measure Theory, Grundlehren Math. Wiss. vol. 153, Springer-Verlag, New York Inc., New York (1969) | MR 257325 | Zbl 0176.00801

[15] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monogr. Math. vol. 80, Birkhäuser Verlag, Basel (1984) | MR 775682 | Zbl 0545.49018

[16] P. Hajłasz, Sobolev spaces on metric-measure spaces, Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces, Paris, 2002, Contemp. Math. vol. 338, Amer. Math. Soc., Providence, RI (2003), 173-218 | MR 2039955 | Zbl 1048.46033

[17] P. Hajłasz, P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 no. 688 (2000) | MR 1683160 | Zbl 0954.46022

[18] H. Hakkarainen, J. Kinnunen, The BV-capacity in metric spaces, Manuscripta Math. 132 no. 1–2 (2010), 51-73 | MR 2609288 | Zbl 1194.28001

[19] H. Hakkarainen, N. Shanmugalingam, Comparisons of relative BV-capacities and Sobolev capacity in metric spaces, Nonlinear Anal. 74 no. 16 (2011), 5525-5543 | MR 2819294 | Zbl 1248.28002

[20] J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext, Springer-Verlag, New York (2001) | MR 1800917 | Zbl 0985.46008

[21] J. Heinonen, P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), 1-61 | MR 1654771 | Zbl 0915.30018

[22] E. Järvenpää, M. Järvenpää, K. Rogovin, S. Rogovin, N. Shanmugalingam, Measurability of equivalence classes and 𝑀𝐸𝐶 p -property in metric spaces, Rev. Mat. Iberoam. 23 no. 3 (2007), 811-830 | MR 2414493 | Zbl 1146.28001

[23] S. Keith, Modulus and the Poincaré inequality on metric measure spaces, Math. Z. 245 no. 2 (2003), 255-292 | MR 2013501 | Zbl 1037.31009

[24] J. Kinnunen, R. Korte, Characterizations of Sobolev inequalities on metric spaces, J. Math. Anal. Appl. 344 no. 2 (2008), 1093-1104 | MR 2426336 | Zbl 1154.46018

[25] J. Kinnunen, R. Korte, N. Shanmugalingam, H. Tuominen, Lebesgue points and capacities via the boxing inequality in metric spaces, Indiana Univ. Math. J. 57 no. 1 (2008), 401-430 | MR 2400262 | Zbl 1146.46018

[26] J. Kinnunen, R. Korte, N. Shanmugalingam, H. Tuominen, A characterization of Newtonian functions with zero boundary values, Calc. Var. Partial Differential Equations 43 no. 3–4 (2012), 507-528 | MR 2875650 | Zbl 1238.31008

[27] J. Kinnunen, R. Korte, N. Shanmugalingam, H. Tuominen, Pointwise properties of functions of bounded variation in metric spaces, preprint, 2011. | MR 3149179

[28] R. Korte, Geometric implications of the Poincaré inequality, Results Math. 50 no. 1–2 (2007), 93-107 | MR 2313133 | Zbl 1146.46019

[29] O. Kurka, Optimal quality of exceptional points for the Lebesgue density theorem, Acta Math. Hungar. 134 no. 3 (2011), 209-268 | MR 2886206 | Zbl 1299.28001

[30] P. Lahti, H. Tuominen, A pointwise characterization of functions of bounded variation on metric spaces, arXiv:1301.6897v1 | MR 3211057 | Zbl 1327.46036

[31] M. Miranda, Functions of bounded variation on “good” metric spaces, J. Math. Pures Appl. (9) 82 no. 8 (2003), 975-1004 | MR 2005202 | Zbl 1109.46030

[32] N. Shanmugalingam, Newtonian spaces: an extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoam. 16 no. 2 (2000), 243-279 | MR 1809341 | Zbl 0974.46038

[33] A. Szenes, Exceptional points for Lebesgue's density theorem on the real line, Adv. Math. 226 no. 1 (2011), 764-778 | MR 2735774 | Zbl 1205.28001

[34] W.P. Ziemer, Weakly Differentiable Functions, Grad. Texts in Math. vol. 120, Springer-Verlag, New York (1989) | MR 1014685 | Zbl 0177.08006