On the isoperimetric problem in Euclidean space with density

Calculus of Variations and Partial Differential Equations

Volumn 31, Issue 1, 2008, Pages 27-46

  Rosales, César ^a   Cañete, Antonio ^a   Bayle, Vincent ^b   Morgan, Frank ^c  

^a UNIVERSITY OF GRANADA   (Spain)

^b INSTITUT FOURIER   (France)

^c WILLIAMS COLLEGE   (United States)

Author keywords

Generalized mean curvature; Isoperimetric problem; Manifolds with density; Stability; Symmetrization

Indexed keywords

EID: 34848888287     PISSN: 09442669     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00526-007-0104-y     Document Type: Article

References (31)

1. Bakry D., Émery M. (1985). Diffusions hypercontractives, Séminaire de probabilités, XIX, 1983/84. Lecture Notes Math. 1123: 177-206
2. Bakry D., Ledoux M. (1996). Lévy-Gromov's isoperimetric inequality for an infinite-dimensional diffusion generator. Invent. Math. 123(2): 259-281
3. Lucas Barbosa J., do Carmo M. (1984). Stability of hypersurfaces with constant mean curvature. Math. Z. 185(3): 339-353
4. Barthe F. (2002). Log-concave and spherical models in isoperimetry. Geom. Funct. Anal. 12(1): 32-55
5. Barthe F., Maurey B. (2000). Some remarks on isoperimetry of Gaussian type. Ann. Inst. H. Poincaré Probab. Stat. 36(4): 419-434
6. Bayle, V.: Propriétés de concavité du profil isopérimétrique et applications. Thèse de Doctorat (2003)
7. Bieberbach L. (1915). Ub̈er eine Extremaleigenschaft des Kreises. J.-ber. Deutsch. Math.-Verein. 24: 247-250
8. Bobkov S. (1996). Extremal properties of half-spaces for log-concave distributions. Ann. Probab. 24: 35-48
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
10. Bobkov, S., Houdré, C.: Some connections between isoperimetric and Sobolev-type inequalities. Mem. Amer. Math. Soc. 129 (1997)
11. Borell C. (1975). The Brunn-Minkoski inequality in Gauss space. Invent. Math. 30(2): 207-216
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)
14. Borell, C.: Analytic and empirical evidences of isoperimetric processes. Probability in Banach spaces 6 (Sandbjerg, 1986), 13-40,
15. Progr. Probab., 20, Birkhäuser Boston, Boston (1990)
16. Carlen E.A., Kerce C. (2001). On the cases of equality in Bobkov's inequality and Gaussian rearrangement. Calc. Var. 13: 1-18
17. Chavel, I.: Eigenvalues in Riemannian Geometry. Pure and Applied Mathematics, vol. 115, Academic, Orlando, (1984)
18. Chavel, I.: Isoperimetric Inequalities. Differential Geometric and Analytic Perspectives. Cambridge Tracts in Mathematics, no. 145, Cambridge University Press, Cambridge (2001)
19. Ehrhard A. (1982). Symétrisation dans l'espace de Gauss. Math. Scand. 53: 281-301
20. Ehrhard A. (1986). Éléments extrémaux pour les inégalités de Brunn-Minkowski gaussiennes. Ann. Inst. H. Poincaré Probab. Stat. 22(2): 149-168
21. Gromov M. (2003). Isoperimetry of waists and concentration of maps. Geom. Funct. Anal. 13: 178-215
22. Hsiang W.Y. (1988). A symmetry theorem on isoperimetric regions. PAM-409, UC Berkeley
23. Morgan F. (1994). Clusters minimizing area plus length of singular curves. Math. Ann. 299: 697-714
24. Morgan F. (2000). Geometric measure theory: a beginner's guide 3rd ed. Academic, San Diego
25. Morgan F. (2003). Regularity of isoperimetric hypersurfaces in Riemannian manifolds. Trans. Am. Math. Soc. 355(12): 5041-5052
26. Morgan F. (2005). Manifolds with density. Notices Am. Math. Soc. 52: 853-858
27. 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
28. 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)
29. Rosenberg H. (1993). Hypersurfaces of constant curvature in space forms. Bull. Sci. Math. 117: 211-239
30. 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)
31. Sudakov, V.N., Tsirel'son, B.S.: Extremal properties of half-spaces for spherically invariant measures. J. Soviet Math. 9-18 (1978)