On the isoperimetric problem in Euclidean space with density. (English) Zbl 1126.49038
Summary: We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive stability conditions, which lead to the conjecture that for a radial log-convex density, balls about the origin are isoperimetric regions. Finally, we prove this conjecture and the uniqueness of minimizers for the density exp\((|x|^2)\) by using symmetrization techniques.
MSC:
| 49Q20 | Variational problems in a geometric measure-theoretic setting |
| 53C17 | Sub-Riemannian geometry |
| 49N60 | Regularity of solutions in optimal control |
Keywords:
Manifolds with density; Isoperimetric problem; Generalized mean curvature; Stability; SymmetrizationReferences:
| [1] | Bakry D., Émery M. (1985). Diffusions hypercontractives, Séminaire de probabilités, XIX, 1983/84. Lecture Notes Math. 1123: 177–206 · doi:10.1007/BFb0075847 |
| [2] | Bakry D., Ledoux M. (1996). Lévy–Gromov’s isoperimetric inequality for an infinite-dimensional diffusion generator. Invent. Math. 123(2): 259–281 · Zbl 0855.58011 |
| [3] | Lucas Barbosa J., do Carmo M. (1984). Stability of hypersurfaces with constant mean curvature. Math. Z. 185(3): 339–353 · doi:10.1007/BF01215045 |
| [4] | Barthe F. (2002). Log-concave and spherical models in isoperimetry. Geom. Funct. Anal. 12(1): 32–55 · Zbl 0999.60017 · doi:10.1007/s00039-002-8235-y |
| [5] | Barthe F., Maurey B. (2000). Some remarks on isoperimetry of Gaussian type. Ann. Inst. H. Poincaré Probab. Stat. 36(4): 419–434 · Zbl 0964.60018 · doi:10.1016/S0246-0203(00)00131-X |
| [6] | Bayle, V.: Propriétés de concavité du profil isopérimétrique et applications. Thèse de Doctorat (2003) |
| [7] | Bieberbach L. (1915). Uber eine Extremaleigenschaft des Kreises. J.-ber. Deutsch. Math.-Verein. 24: 247–250 · JFM 45.0623.01 |
| [8] | Bobkov S. (1996). Extremal properties of half-spaces for log-concave distributions. Ann. Probab. 24: 35–48 · Zbl 0859.60048 · doi:10.1214/aop/1042644706 |
| [9] | Bobkov S. (1997). An isoperimetric inequality on the discrete cube and an elementary proof of the isoperimetric inequality in Gauss space. Ann. Probab. 25(1): 206–214 · Zbl 0883.60031 · doi:10.1214/aop/1024404285 |
| [10] | Bobkov, S., Houdré, C.: Some connections between isoperimetric and Sobolev-type inequalities. Mem. Amer. Math. Soc. 129 (1997) · Zbl 0886.49033 |
| [11] | Borell C. (1975). The Brunn-Minkoski inequality in Gauss space. Invent. Math. 30(2): 207–216 · doi:10.1007/BF01425510 |
| [12] | Borell, C.: The Ornstein–Uhlenbeck velocity process in backward time and isoperimetry. Chalmers University of Technology 1986-03/ISSN 0347-2809 (preprint) |
| [13] | Borell, C.: Intrinsic bounds for some real-valued stationary random functions. Lecture Notes in Math. 1153, 72–95, Springer, Berlin, (1985) · Zbl 0607.60067 |
| [14] | Borell, C.: Analytic and empirical evidences of isoperimetric processes. Probability in Banach spaces 6 (Sandbjerg, 1986), 13–40, Progr. Probab., 20, Birkhäuser Boston, Boston (1990) |
| [15] | Carlen E.A., Kerce C. (2001). On the cases of equality in Bobkov’s inequality and Gaussian rearrangement. Calc. Var. 13: 1–18 · Zbl 1009.49029 · doi:10.1007/PL00009921 |
| [16] | Chavel, I.: Eigenvalues in Riemannian Geometry. Pure and Applied Mathematics, vol. 115, Academic, Orlando, (1984) · Zbl 0551.53001 |
| [17] | Chavel, I.: Isoperimetric Inequalities. Differential Geometric and Analytic Perspectives. Cambridge Tracts in Mathematics, no. 145, Cambridge University Press, Cambridge (2001) · Zbl 0988.51019 |
| [18] | Ehrhard A. (1982). Symétrisation dans l’espace de Gauss. Math. Scand. 53: 281–301 · Zbl 0542.60003 |
| [19] | Ehrhard A. (1986). Éléments extrémaux pour les inégalités de Brunn-Minkowski gaussiennes. Ann. Inst. H. Poincaré Probab. Stat. 22(2): 149–168 · Zbl 0595.60020 |
| [20] | Gromov M. (2003). Isoperimetry of waists and concentration of maps. Geom. Funct. Anal. 13: 178–215 · Zbl 1122.20021 · doi:10.1007/s000390300002 |
| [21] | Hsiang W.Y. (1988). A symmetry theorem on isoperimetric regions. PAM-409, UC Berkeley |
| [22] | Morgan F. (1994). Clusters minimizing area plus length of singular curves. Math. Ann. 299: 697–714 · Zbl 0805.49025 · doi:10.1007/BF01459806 |
| [23] | Morgan F. (2000). Geometric measure theory: a beginner’s guide 3rd ed. Academic, San Diego · Zbl 0974.49025 |
| [24] | Morgan F. (2003). Regularity of isoperimetric hypersurfaces in Riemannian manifolds. Trans. Am. Math. Soc. 355(12): 5041–5052 · Zbl 1063.49031 · doi:10.1090/S0002-9947-03-03061-7 |
| [25] | Morgan F. (2005). Manifolds with density. Notices Am. Math. Soc. 52: 853–858 · Zbl 1118.53022 |
| [26] | Ritoré M., Rosales C. (2004). Existence and characterization of regions minimizing perimeter under a volume constraint inside Euclidean cones. Trans. Am. Math. Soc. 356(11): 4601–4622 · Zbl 1057.53023 · doi:10.1090/S0002-9947-04-03537-8 |
| [27] | Ros, A.: The isoperimetric problem. Global Theory of Minimal Surfaces. In: Hoffman D Proceedings of Clay Mathematics Institute 2001 Summer School, MSRI. Amer. Math. Soc. 175–209 (2005) · Zbl 1125.49034 |
| [28] | Rosenberg H. (1993). Hypersurfaces of constant curvature in space forms. Bull. Sci. Math. 117: 211–239 · Zbl 0787.53046 |
| [29] | Simon, L.: Lectures on geometric measure theory. In: Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University Centre for Mathematical Analysis, Canberra (1983) · Zbl 0546.49019 |
| [30] | Sudakov, V.N., Tsirel’son, B.S.: Extremal properties of half-spaces for spherically invariant measures. J. Soviet Math. 9–18 (1978) · Zbl 0395.28007 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
