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On the Cheeger sets in strips and non-convex domains. (English) Zbl 1337.49074

The Cheeger problem is an isoperimetric-like problem. It has connections to Riemannian geometry, calculus of variations and eigenvalues problem. For a domain \(\Omega \subseteq \mathbb{R}^n\) one looks for an estimate of the Cheeger constant, defined as \(h(\Omega) =\inf{\{{ {P(F)}\over {|F|}}}: F\subseteq \Omega, |F|>0\}\), where \(|F|\) and \(P(F)\) denote, respectively, the volume and the perimeter of a Borel set \(F\subseteq\mathbb{R}^n\). A set \(F\subseteq \Omega\) which realizes the above infimum is called a Cheeger set in \(\Omega\). After stating some basic properties of the Cheeger sets, the authors obtain the continuity of the Cheeger constant. Then, they discuss the Cheeger problem in convex domains obtaining a certain uniqueness and convexity result. They obtain some further results about Cheeger sets in \(\mathbb{R}^2\). Some characterizations of Cheeger sets in planar strips are given. Among these characterizations Theorem 3.2 is notable. Let \(S\) be also an open strip of length \(L\) and width 2. Then, \(h(S)=1+{{\pi}\over{2L}}+O(L^{-2})\) as \(L\to +\infty\).
Then the authors prove some simple technical relations between the perimeter and the area of a strip and the length of the corresponding spinal curve. The final result is Theorem 3.3., asserting the existence and the uniqueness of a Cheeger set for a certain open strip \(S\). Next, the authors provide some planar examples.

MSC:

49Q10 Optimization of shapes other than minimal surfaces
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
35P15 Estimates of eigenvalues in context of PDEs
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[1] Alter, F; Caselles, V, Uniqueness of the Cheeger set of a convex body, Nonlinear Anal., 70, 32-44, (2009) · Zbl 1167.52005
[2] Ambrosio, L; Colesanti, A; Villa, E, Outer Minkowski content for some classes of closed sets, Math. Ann., 342, 727-748, (2008) · Zbl 1152.28005
[3] Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems, vol. 254. Clarendon Press, Oxford (2000) · Zbl 0957.49001
[4] Buttazzo, G; Carlier, G; Comte, M, On the selection of maximal Cheeger sets, Differ. Integral Equ., 20, 991-1004, (2007) · Zbl 1212.49019
[5] Carlier, G; Comte, M; Peyré, G, Approximation of maximal Cheeger sets by projection. M2AN, Math. Model. Numer. Anal., 43, 139-150, (2009) · Zbl 1161.65046
[6] Caselles, V; Chambolle, A; Moll, S; Novaga, M, A characterization of convex calibrable sets in \(\mathbb{R}^N\) with respect to anisotropic norms, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25, 803-832, (2008) · Zbl 1144.52002
[7] Caselles, V; Chambolle, A; Novaga, M, Some remarks on uniqueness and regularity of Cheeger sets, Rend. Semin. Mat. Univ. Padova, 123, 191-201, (2010) · Zbl 1198.49042
[8] Caselles, V; Facciolo, G; Meinhardt, E, Anisotropic Cheeger sets and applications, SIAM J. Imaging Sci., 2, 1211-1254, (2009) · Zbl 1193.49051
[9] Caselles, V; Miranda, MJ; Novaga, M, Total variation and Cheeger sets in Gauss space, J. Funct. Anal., 259, 1491-1516, (2010) · Zbl 1195.49054
[10] De Giorgi, E.: Selected Papers. Springer-Verlag, Berlin (2006)
[11] Duclos, P; Exner, P, Curvature-induced bound states in quantum waveguides in two and three dimensions, Rev. Math. Phys., 7, 73-102, (1995) · Zbl 0837.35037
[12] Federer, H, Curvature measures, Trans. Amer. Math. Soc., 93, 418-491, (1959) · Zbl 0089.38402
[13] Federer, H.: Geometric Measure Theory, Volume 153 of Die Grundlehren der Mathematischen Wissenschaften. Springer-Verlag New York Inc., New York (1969)
[14] Ionescu, IR; Lachand-Robert, T, Generalized Cheeger sets related to landslides, Calc. Var. Partial Differ. Equ., 23, 227-249, (2005) · Zbl 1062.49036
[15] Kawohl, B; Fridman, V, Isoperimetric estimates for the first eigenvalue of the \(p\)-Laplace operator and the Cheeger constant, Comment. Math. Univ. Carolin., 44, 659-667, (2003) · Zbl 1105.35029
[16] Kawohl, B; Lachand-Robert, T, Characterization of Cheeger sets for convex subsets of the plane, Pac. J. Math., 225, 103-118, (2006) · Zbl 1133.52002
[17] Krejčiřík, D; Kříz, J, On the spectrum of curved planar waveguides, Publ. Res. Inst. Math. Sci., 41, 757-791, (2005) · Zbl 1113.35143
[18] Krejčiřík, D; Pratelli, A, The Cheeger constant of curved strips, Pac. J. Math., 254, 309-333, (2011) · Zbl 1247.28003
[19] Leonardi, GP; Pratelli, A (ed.); Leugering, G (ed.), An overview on the Cheeger problem, No. 166, 117-139, (2015), Switzerland · Zbl 1329.49088
[20] Maggi, F.: Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory. Cambridge Studies in Advanced Mathematics, vol. 135. Cambridge University Press, Cambridge (2012) · Zbl 1255.49074
[21] Parini, E.: Cheeger sets in the nonconvex case. Università degli Studi di Milano, Tesi di Laurea Magistrale (2006) · JFM 65.0796.01
[22] Parini, E.: Asymptotic behaviour of higher eigenfunctions of the p-Laplacian as p goes to 1. PhD thesis, Universität zu Köln (2009) · Zbl 1336.35004
[23] Pratelli, A., Saracco, G.: On the generalized Cheeger problem and an application to 2d rectangles (preprint) (2014) · Zbl 1161.65046
[24] Simon, L, A strict maximum principle for area minimizing hypersurfaces, J. Differ. Geom., 26, 327-335, (1987) · Zbl 0625.53052
[25] Steiner, J.: Über parallele flächen. Monatsber. Preuss. Akad. Wiss. pp. 114-118 (1840)
[26] Strang, G, Maximum flows and minimum cuts in the plane, J. Global Optim., 47, 527-535, (2010) · Zbl 1207.49042
[27] Stredulinsky, E; Ziemer, WP, Area minimizing sets subject to a volume constraint in a convex set, J. Geom. Anal., 7, 653-677, (1997) · Zbl 0940.49025
[28] Tamanini, I.: Regularity results for almost minimal oriented hypersurfaces in \(\mathbb{R}^N\). Quaderni del Dipartimento di Matematica dell’Università di Lecce. http://cvgmt.sns.it/paper/1807/ (1984) · Zbl 1191.35007
[29] Weyl, H, On the volume of tubes, Am. J. Math., 61, 461-472, (1939) · Zbl 0021.35503
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