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Uniqueness of the Cheeger set of a convex body. (English) Zbl 1167.52005
A Cheeger set of a nonempty open bounded subset \(\Omega\subset{\mathbb R}^N\) is any set \(G\subseteq\Omega\) minimizing the ratio \(P(F)/|F|\), \(F\subseteq\Omega\), where \(|F|\) denotes the \(N\)-dimensional volume of \(F\) and \(P(F)\) the measure of its boundary.
In this paper the authors show that convexity of the set \(\Omega\) implies the uniqueness of the Cheeger set, which will be, in addition, convex and of class \(C^{1,1}\). Due to former results the main point here is the uniqueness, which has been known only in the planar case or under additional assumptions. In the planar case, the unique Cheeger set has the nice geometric property to been just the outer parallel body of a certain inner parallel body of \(\Omega\) [B. Kawohl and T. Lachand-Robert, Pac. J. Math. 225, No. 1, 103–118 (2006; Zbl 1133.52002)].

MSC:
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
49Q20 Variational problems in a geometric measure-theoretic setting
35J60 Nonlinear elliptic equations
52A38 Length, area, volume and convex sets (aspects of convex geometry)
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[1] Almgren, F.J.; Taylor, J.E.; Wang, L.-H., Curvature-driven flows: A variational approach, SIAM J. control optim., 31, 2, 387-438, (1993) · Zbl 0783.35002
[2] Alter, F.; Caselles, V.; Chambolle, A., A characterization of convex calibrable sets in \(\mathbb{R}^N\), Math. ann., 332, 2, 329-366, (2005) · Zbl 1108.35073
[3] Alter, F.; Caselles, V.; Chambolle, A., Evolution of characteristic functions of convex sets in the plane by the minimizing total variation flow, Interfaces free bound., 7, 1, 29-53, (2005) · Zbl 1084.49003
[4] Alvarez, O.; Lasry, J.M.; Lions, P.L., Convex viscosity solutions and state constraints, J. math. pures appl. (9), 76, 3, 265-288, (1997) · Zbl 0890.49013
[5] Bangert, V., Convex hypersurfaces with bounded first Mean curvature measure, Calc. var., 8, 259-278, (1999) · Zbl 0960.53007
[6] Barozzi, E., The curvature of a set with finite area, Atti accad. naz. lincei cl. sci. fis. mat. natur. rend. lincei (9) mat. appl., 5, 2, 149-159, (1994) · Zbl 0809.49038
[7] Bellettini, G.; Caselles, V.; Novaga, M., The total variation flow in \(\mathbb{R}^N\), J. differential equations, 184, 2, 475-525, (2002) · Zbl 1036.35099
[8] Bellettini, G.; Caselles, V.; Chambolle, A.; Novaga, M., Crystalline Mean curvature evolution of convex sets, Arch. ration. math. mech., 179, 1, 109-152, (2006) · Zbl 1148.53049
[9] G. Buttazzo, G. Carlier, M. Comte, On the selection of maximal Cheeger sets, preprint, available at http://cvgmt.sns.it/ · Zbl 1212.49019
[10] Carlier, G.; Comte, M., On a weighted total variation minimization problem, J. funct. anal., 250, 214-226, (2007) · Zbl 1120.49011
[11] Caselles, V.; Chambolle, A.; Novaga, M., Uniqueness of the Cheeger set of a convex body, Pacific J. math., 232, 1, 77-90, (2007) · Zbl 1221.35171
[12] Cheeger, J., A lower bound for the smallest eigenvalue of the Laplacian, (), 195-199 · Zbl 0212.44903
[13] Chow, B., Deforming convex hypersurfaces by the square root of the scalar curvature, Invent. math., 87, 63-82, (1987) · Zbl 0608.53005
[14] Evans, L.C.; Spruck, J., Motion of level sets by Mean curvature I, J. differential geom., 33, 635-681, (1991) · Zbl 0726.53029
[15] Evans, L.C.; Spruck, J., Motion of level sets by Mean curvature III, J. geom. anal., 2, 2, 121-150, (1992) · Zbl 0768.53003
[16] F.R. Gantmacher, Théorie des matrices, Éditions Jacques Gabay, 1990
[17] Giusti, E., On the equation of surfaces of prescribed Mean curvature. existence and uniqueness without boundary conditions, Invent. math., 46, 2, 111-137, (1978) · Zbl 0381.35035
[18] Giusti, E., Minimal surfaces and functions of bounded variation, (1983), Birkhaüser
[19] Gonzalez, E.H.A.; Massari, U., Variational Mean curvatures, Rend. sem. mat. univ. politec. Torino, 52, 1, 1-28, (1994) · Zbl 0819.49025
[20] Gonzalez, E.H.A.; Massari, U.; Tamanini, I., Minimal boundaries enclosing a given volume, Manuscripta math., 34, 381-395, (1981) · Zbl 0481.49035
[21] Gonzalez, E.H.A.; Massari, U.; Tamanini, I., On the regularity of sets minimizing perimeter with a volume constraint, Indiana univ. math. J., 32, 25-37, (1983) · Zbl 0486.49024
[22] Grieser, D., The first eigenvalue of the Laplacian, isoperimetric constants, and the MAX-flow MIN-cut theorem, Arch. math. (basel), 87, 1, 75-85, (2006) · Zbl 1105.35062
[23] Huang, W.H., Superharmonicity of curvatures for surfaces of constant Mean curvature, Pacific J. math., 152, 2, 291-318, (1992) · Zbl 0767.53040
[24] Huisken, G., Flow by Mean curvature of convex surfaces into spheres, J. differential geom., 20, 237-266, (1984) · Zbl 0556.53001
[25] Ionescu, I.R.; Lachand-Robert, T., Generalized Cheeger sets related to landslides, Calc. var. partial differential equations, 23, 2, 227-249, (2005) · Zbl 1062.49036
[26] Kawohl, B., On a family of torsional creep problems, J. reine angew. math., 410, 1-22, (1990) · Zbl 0701.35015
[27] Kawohl, B.; Fridman, V., Isoperimetric estimates for the first eigenvalue of the \(p\)-Laplace operator and the Cheeger constant, Comment. math. univ. carolin., 44, 4, 659-667, (2003) · Zbl 1105.35029
[28] Kawohl, B.; Kutev, N., Global behaviour of solutions to a parabolic Mean curvature equation, Differential integral equations, 8, 1923-1946, (1995) · Zbl 0844.35050
[29] Kawohl, B.; Lachand-Robert, T., Characterization of Cheeger sets for convex subsets of the plane, Pacific J. math., 225, 1, 103-118, (2006) · Zbl 1133.52002
[30] Kawohl, B.; Novaga, M., The \(p\)-Laplace eigenvalue problem as \(p \rightarrow 1\) and Cheeger sets in a Finsler metric, J. convex anal., 15, 3, (2008) · Zbl 1186.35115
[31] Lachand-Robert, T.; Oudet, E., Minimizing within convex bodies using a convex hull method, SIAM J. optim., 16, 2, 368-379, (2005) · Zbl 1104.65056
[32] Lefton, L.; Wei, D., Numerical approximation of the first eigenvalue of the \(p\)-Laplacian using finite elements and the penalty method, Numer. funct. anal. optim., 18, 389-399, (1997) · Zbl 0884.65103
[33] Marcellini, P.; Miller, K., Asymptotic growth for the parabolic equation of prescribed Mean curvature, J. differential equations, 51, 3, 326-358, (1984) · Zbl 0545.35044
[34] Schneider, R., Convex bodies: the brunn-Minkowski theory, () · Zbl 1287.52001
[35] Simon, L., Lectures on geometric measure theory, () · Zbl 0546.49019
[36] Strang, G., Maximal flow through a domain, Math. program., 26, 2, 123-143, (1983) · Zbl 0513.90026
[37] Stredulinsky, E.; Ziemer, W.P., Area minimizing sets subject to a volume constraint in a convex set, J. geom. anal., 7, 4, 653-677, (1997) · Zbl 0940.49025
[38] Tamanini, I., Boundaries of cacciopoli sets with Hölder-continuous normal vector, J. reine angew. math., 334, 27-39, (1982) · Zbl 0479.49028
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