Isoperimetric and Stable Sets for Log-Concave Perturbations of Gaussian Measures

César Rosales

Analysis and Geometry in Metric Spaces (2014)

  • Volume: 2, Issue: 1, page 328-358, electronic only
  • ISSN: 2299-3274

Abstract

top
Let be an open half-space or slab in ℝn+1 endowed with a perturbation of the Gaussian measure of the form f (p) := exp(ω(p) − c|p|2), where c > 0 and ω is a smooth concave function depending only on the signed distance from the linear hyperplane parallel to ∂ Ω. In this work we follow a variational approach to show that half-spaces perpendicular to ∂ Ω uniquely minimize the weighted perimeter in Ω among sets enclosing the same weighted volume. The main ingredient of the proof is the characterization of half-spaces parallel or perpendicular to ∂ Ω as the unique stable sets with small singular set and null weighted capacity. Our methods also apply for = ℝn+1, which produces in particular the classification of stable sets in Gauss space and a new proof of the Gaussian isoperimetric inequality. Finally, we use optimal transport to study the weighted minimizers when the perturbation term ω is concave and possibly non-smooth.

How to cite

top

César Rosales. "Isoperimetric and Stable Sets for Log-Concave Perturbations of Gaussian Measures." Analysis and Geometry in Metric Spaces 2.1 (2014): 328-358, electronic only. <http://eudml.org/doc/268885>.

@article{CésarRosales2014,
abstract = {Let be an open half-space or slab in ℝn+1 endowed with a perturbation of the Gaussian measure of the form f (p) := exp(ω(p) − c|p|2), where c > 0 and ω is a smooth concave function depending only on the signed distance from the linear hyperplane parallel to ∂ Ω. In this work we follow a variational approach to show that half-spaces perpendicular to ∂ Ω uniquely minimize the weighted perimeter in Ω among sets enclosing the same weighted volume. The main ingredient of the proof is the characterization of half-spaces parallel or perpendicular to ∂ Ω as the unique stable sets with small singular set and null weighted capacity. Our methods also apply for = ℝn+1, which produces in particular the classification of stable sets in Gauss space and a new proof of the Gaussian isoperimetric inequality. Finally, we use optimal transport to study the weighted minimizers when the perturbation term ω is concave and possibly non-smooth.},
author = {César Rosales},
journal = {Analysis and Geometry in Metric Spaces},
keywords = {Log-concave densities; Gaussian measures; isoperimetric problems; stable sets; free boundary hypersurfaces; log-concave densities},
language = {eng},
number = {1},
pages = {328-358, electronic only},
title = {Isoperimetric and Stable Sets for Log-Concave Perturbations of Gaussian Measures},
url = {http://eudml.org/doc/268885},
volume = {2},
year = {2014},
}

TY - JOUR
AU - César Rosales
TI - Isoperimetric and Stable Sets for Log-Concave Perturbations of Gaussian Measures
JO - Analysis and Geometry in Metric Spaces
PY - 2014
VL - 2
IS - 1
SP - 328
EP - 358, electronic only
AB - Let be an open half-space or slab in ℝn+1 endowed with a perturbation of the Gaussian measure of the form f (p) := exp(ω(p) − c|p|2), where c > 0 and ω is a smooth concave function depending only on the signed distance from the linear hyperplane parallel to ∂ Ω. In this work we follow a variational approach to show that half-spaces perpendicular to ∂ Ω uniquely minimize the weighted perimeter in Ω among sets enclosing the same weighted volume. The main ingredient of the proof is the characterization of half-spaces parallel or perpendicular to ∂ Ω as the unique stable sets with small singular set and null weighted capacity. Our methods also apply for = ℝn+1, which produces in particular the classification of stable sets in Gauss space and a new proof of the Gaussian isoperimetric inequality. Finally, we use optimal transport to study the weighted minimizers when the perturbation term ω is concave and possibly non-smooth.
LA - eng
KW - Log-concave densities; Gaussian measures; isoperimetric problems; stable sets; free boundary hypersurfaces; log-concave densities
UR - http://eudml.org/doc/268885
ER -

References

top
  1. [1] E. Adams, I. Corwin, D. Davis, M. Lee, and R. Visocchi, Isoperimetric regions in Gauss sectors, Rose-Hulman Und. Math. J. 8 (2007), no. 1. 
  2. [2] L. Ambrosio, Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces, Adv.Math. 159 (2001), no. 1, 51–67. MR 1823840 (2002b:31002) Zbl1002.28004
  3. [3] L. Ambrosio, N. Fusco, and D. Pallara, Functions of bounded variation and free discontinuity problems, OxfordMathematical Monographs, The Clarendon Press Oxford University Press, New York, 2000. MR 1857292 (2003a:49002) Zbl0957.49001
  4. [4] D. Bakry and M. Émery, Diffusions hypercontractives, Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 177–206. MR MR889476 (88j:60131) 
  5. [5] D. Bakry and M. Ledoux, Lévy-Gromov’s isoperimetric inequality for an infinite-dimensional diffusion generator, Invent. Math. 123 (1996), no. 2, 259–281. MR MR1374200 (97c:58162) Zbl0855.58011
  6. [6] D. Bakry and Z. Qian, Some new results on eigenvectors via dimension, diameter, and Ricci curvature, Adv.Math. 155 (2000), no. 1, 98–153. MR 1789850 (2002g:58048) Zbl0980.58020
  7. [7] A. Baldi, Weighted BV functions, Houston J. Math. 27 (2001), no. 3, 683–705. MR 1864805 (2002j:46045) 
  8. [8] J. L. Barbosa, M. P. do Carmo, and J. Eschenburg, Stability of hypersurfaces of constant mean curvature in Riemannian manifolds, Math. Z. 197 (1988), no. 1, 123–138. MR MR917854 (88m:53109) Zbl0653.53045
  9. [9] M. Barchiesi, A. Brancolini, and V. Julin, Sharp dimension free quantitative estimates for the Gaussian isoperimetric inequality, arXiv:1409.2106, September 2014. 
  10. [10] F. Barthe, C. Bianchini, and A. Colesanti, Isoperimetry and stability of hyperplanes for product probability measures, Ann. Mat. Pura Appl. (4) 192 (2013), no. 2, 165–190. MR 3035134 Zbl1267.53014
  11. [11] F. Barthe and B. Maurey, Some remarks on isoperimetry of Gaussian type, Ann. Inst. H. Poincaré Probab. Statist. 36 (2000), no. 4, 419–434. MR MR1785389 (2001k:60055) Zbl0964.60018
  12. [12] V. Bayle, Propriétés de concavité du profil isopérimétrique et applications, Ph.D. thesis, Institut Fourier (Grenoble), 2003. 
  13. [13] , A differential inequality for the isoperimetric profile, Int. Math. Res. Not. (2004), no. 7, 311–342. MR 2041647 (2005a:53050) Zbl1080.53026
  14. [14] V. Bayle and C. Rosales, Some isoperimetric comparison theorems for convex bodies in Riemannian manifolds, Indiana Univ. Math. J. 54 (2005), no. 5, 1371–1394. MR 2177105 (2006f:53040) Zbl1085.53025
  15. [15] G. Bellettini, G. Bouchitté, and I. Fragalà, BV functions with respect to a measure and relaxation of metric integral functionals, J. Convex Anal. 6 (1999), no. 2, 349–366. MR 1736243 (2000k:49016) Zbl0959.49015
  16. [16] S. G. Bobkov, An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space, Ann. Probab. 25 (1997), no. 1, 206–214. MR MR1428506 (98g:60033) Zbl0883.60031
  17. [17] , Perturbations in the Gaussian isoperimetric inequality, J. Math. Sci. (N. Y.) 166 (2010), no. 3, 225–238, Problems in mathematical analysis. No. 45. MR 2839030 (2012m:60050) 
  18. [18] S. G. Bobkov and K. Udre, Characterization of Gaussian measures in terms of the isoperimetric property of half-spaces, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 228 (1996), no. Veroyatn. i Stat. 1, 31–38, 356. MR 1449845 (98e:60056) 
  19. [19] C. Borell, The Brunn-Minkowski inequality in Gauss space, Invent. Math. 30 (1975), no. 2, 207–216. MR MR0399402 (53 #3246) Zbl0292.60004
  20. [20] F. Brock, F. Chiacchio, and A. Mercaldo, A class of degenerate elliptic equations and a Dido’s problem with respect to a measure, J. Math. Anal. Appl. 348 (2008), no. 1, 356–365. MR 2449353 (2010h:35146) Zbl1156.35048
  21. [21] X. Cabré, X. Ros-Oton, and J. Serra, Sharp isoperimetric inequalities via the ABP method, arXiv:1304.1724v3, April 2013. Zbl1293.46018
  22. [22] L. A. Caffarelli, Monotonicity properties of optimal transportation and the FKG and related inequalities, Comm. Math. Phys. 214 (2000), no. 3, 547–563. MR 1800860 (2002c:60029) Zbl0978.60107
  23. [23] A. Cañete, M. Miranda, and D. Vittone, Some isoperimetric problems in planes with density, J. Geom. Anal. 20 (2010), no. 2, 243–290. MR 2579510 (2011a:49102) Zbl1193.49050
  24. [24] A. Cañete and C. Rosales, Compact stable hypersurfaces with free boundary in convex solid cones with homogeneous densities, Calc. Var. Partial Differential Equations 51 (2014), no. 3-4, 887–913. MR 3268875 Zbl1317.53006
  25. [25] E. A. Carlen and C. Kerce, On the cases of equality in Bobkov’s inequality and Gaussian rearrangement, Calc. Var. Partial Differential Equations 13 (2001), no. 1, 1–18. MR MR1854254 (2002f:26016) Zbl1009.49029
  26. [26] K. Castro and C. Rosales, Free boundary stable hypersurfaces in manifolds with density and rigidity results, J. Geom. Phys. 79 (2014), 14–28. Zbl1284.53052
  27. [27] G. R. Chambers, Proof of the log-convex density conjecture, arXiv:1311.4012v2, December 2013. 
  28. [28] A. Cianchi, N. Fusco, F. Maggi, and A. Pratelli, On the isoperimetric deficit in Gauss space, Amer. J. Math. 133 (2011), no. 1, 131–186. MR 2752937 (2012b:28007) Zbl1219.28005
  29. [29] E. Cinti and A. Pratelli, The " −"fi property, the boundedness of isoperimetric sets in RN with density, and some applications, arXiv:1209.3624, September 2012. 
  30. [30] T. H. Doan, Some calibrated surfaces in manifolds with density, J. Geom. Phys. 61 (2011), no. 8, 1625–1629. MR 2802497 (2012e:53091) Zbl1225.53019
  31. [31] A. Ehrhard, Symétrisation dans l’espace de Gauss, Math. Scand. 53 (1983), no. 2, 281–301. MR MR745081 (85f:60058) Zbl0542.60003
  32. [32] L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, Studies in AdvancedMathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660 (93f:28001) 
  33. [33] A. Figalli and F. Maggi, On the isoperimetric problem for radial log-convex densities, Calc. Var. Partial Differential Equations 48 (2013), no. 3-4, 447–489. MR 3116018 Zbl1307.49046
  34. [34] N. Fusco, F.Maggi, and A. Pratelli, On the isoperimetric problemwith respect to a mixed Euclidean-Gaussian density, J. Funct. Anal. 260 (2011), no. 12, 3678–3717. MR 2781973 (2012c:49095) Zbl1222.49058
  35. [35] E. Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984. MR 775682 (87a:58041) Zbl0545.49018
  36. [36] E. Gonzalez, U. Massari, and I. Tamanini, On the regularity of boundaries of sets minimizing perimeter with a volume constraint, Indiana Univ. Math. J. 32 (1983), no. 1, 25–37. MR 684753 (84d:49043) Zbl0486.49024
  37. [37] A. Grigor’yan, Isoperimetric inequalities and capacities on Riemannian manifolds, The Maz’ya anniversary collection, Vol. 1 (Rostock, 1998), Oper. Theory Adv. Appl., vol. 109, Birkhäuser, Basel, 1999, pp. 139–153. MR 1747869 (2002a:31009) 
  38. [38] A. Grigor’yan and J. Masamune, Parabolicity and stochastic completeness of manifolds in terms of the Green formula, J. Math. Pures Appl. (9) 100 (2013), no. 5, 607–632. MR 3115827 Zbl06448858
  39. [39] M. Gromov, Isoperimetry of waists and concentration of maps, Geom. Funct. Anal. 13 (2003), no. 1, 178–215. MR MR1978494 (2004m:53073) Zbl1044.46057
  40. [40] M. Grüter, Boundary regularity for solutions of a partitioning problem, Arch. Rational Mech. Anal. 97 (1987), no. 3, 261–270. MR 862549 (87k:49050) Zbl0613.49029
  41. [41] M. Grüter and J. Jost, Allard type regularity results for varifolds with free boundaries, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13 (1986), no. 1, 129–169. MR 863638 (89d:49048) Zbl0615.49018
  42. [42] S. Howe, The log-convex density conjecture and vertical surface area in warped products, arXiv:1107.4402, July 2011. Zbl1326.49077
  43. [43] Y.-H. Kim and E. Milman, A generalization of Caffarelli’s contraction theorem via (reverse) heat flow, Math. Ann. 354 (2012), no. 3, 827–862. MR 2983070 Zbl1257.35101
  44. [44] A. V. Kolesnikov and E. Milman, Poincaré and Brunn-Minkowski inequalities on weighted Riemannian manifolds with boundary, arXiv:1310.2526v4, September 2014. 
  45. [45] M. Ledoux, Isoperimetry and Gaussian analysis, Lectures on probability theory and statistics (Saint-Flour, 1994), Lecture Notes in Math., vol. 1648, Springer, Berlin, 1996, pp. 165–294. MR 1600888 (99h:60002) 
  46. [46] , A short proof of the Gaussian isoperimetric inequality, High dimensional probability (Oberwolfach, 1996), Progr. Probab., vol. 43, Birkhäuser, Basel, 1998, pp. 229–232. MR 1652328 (99j:60027) 
  47. [47] , The geometry ofMarkov diffusion generators, Ann. Fac. Sci. ToulouseMath. (6) 9 (2000), no. 2, 305–366, Probability theory. MR 1813804 (2002a:58045) 
  48. [48] M. Lee, Isoperimetric regions in surfaces and in surfaces with density, Rose-Hulman Und. Math. J. 7 (2006), no. 2. 
  49. [49] G. P. Leonardi and S. Rigot, Isoperimetric sets on Carnot groups, Houston J. Math. 29 (2003), no. 3, 609–637 (electronic). MR MR2000099 (2004d:28008) Zbl1039.49037
  50. [50] A. Lichnerowicz, Variétés riemanniennes à tenseur C non négatif, C. R. Acad. Sci. Paris Sér. A-B 271 (1970), A650–A653. MR 0268812 (42 #3709) Zbl0208.50003
  51. [51] , Variétés kählériennes à première classe de Chern non negative et variétés riemanniennes à courbure de Ricci généralisée non negative, J. Differential Geom. 6 (1971/72), 47–94. MR 0300228 (45 #9274) Zbl0231.53063
  52. [52] F. Maggi, Sets of finite perimeter and geometric variational problems, Cambridge Studies in Advanced Mathematics, vol. 135, Cambridge University Press, Cambridge, 2012, An introduction to geometric measure theory. MR 2976521 Zbl1255.49074
  53. [53] M. McGonagle and J. Ross, The hyperplane is the only stable, smooth solution to the isoperimetric problem in Gaussian space, arXiv:1307.7088, July 2013. Zbl1325.53079
  54. [54] E. Milman, Sharp isoperimetric inequalities and model spaces for curvature-dimension-diameter condition, to appear in J. Eur. Math. Soc., arXiv:1108.4609v3. 
  55. [55] , A proof of Bobkov’s spectral bound for convex domains via Gaussian fitting and free energy estimation, Analysis and geometry of metric measure spaces, CRM Proc. Lecture Notes, vol. 56, Amer. Math. Soc., Providence, RI, 2013, pp. 181–196. MR 3060503 Zbl1275.60024
  56. [56] E. Milman and L. Rotem, Complemented Brunn-Minkowski inequalities and isoperimetry for homogeneous and nonhomogeneous measures, Adv. Math. 262 (2014), 867–908. MR 3228444 Zbl1311.52008
  57. [57] M. Miranda, Functions of bounded variation on “good” metric spaces, J. Math. Pures Appl. (9) 82 (2003), no. 8, 975–1004. MR 2005202 (2004k:46038) Zbl1109.46030
  58. [58] F. Morgan, Regularity of isoperimetric hypersurfaces in Riemannian manifolds, Trans. Amer. Math. Soc. 355 (2003), no. 12, 5041–5052. MR 1997594 (2004j:49066) Zbl1063.49031
  59. [59] , Manifolds with density, Notices Amer. Math. Soc. 52 (2005), no. 8, 853–858. MR MR2161354 (2006g:53044) Zbl1118.53022
  60. [60] , Geometric measure theory. A beginner’s guide, fourth ed., Elsevier/Academic Press, Amsterdam, 2009. MR 2455580 (2009i:49001) 
  61. [61] , The log-convex density conjecture, Concentration, functional inequalities and isoperimetry, Contemp. Math., vol. 545, Amer. Math. Soc., Providence, RI, 2011, pp. 209–211. MR 2858534 
  62. [62] F. Morgan and D. L. Johnson, Some sharp isoperimetric theorems for Riemannianmanifolds, Indiana Univ.Math. J. 49 (2000), no. 3, 1017–1041. MR 1803220 (2002e:53043) Zbl1021.53020
  63. [63] F. Morgan and A. Pratelli, Existence of isoperimetric regions in Rn with density, Ann. Global Anal. Geom. 43 (2013), no. 4, 331–365. MR 3038539 Zbl1263.49049
  64. [64] F. Morgan and M. Ritoré, Isoperimetric regions in cones, Trans. Amer.Math. Soc. 354 (2002), no. 6, 2327–2339. MR 1885654 (2003a:53089) Zbl0988.53028
  65. [65] M. Ritoré and C. Rosales, Existence and characterization of regions minimizing perimeter under a volume constraint inside Euclidean cones, Trans. Amer. Math. Soc. 356 (2004), no. 11, 4601–4622. MR 2067135 (2005g:49076) Zbl1057.53023
  66. [66] A. Ros, The isoperimetric problem, Global theory of minimal surfaces, ClayMath. Proc., vol. 2, Amer.Math. Soc., Providence, RI, 2005, pp. 175–209. MR MR2167260 (2006e:53023) Zbl1125.49034
  67. [67] C. Rosales, A. Cañete, V. Bayle, and F. Morgan, On the isoperimetric problem in Euclidean space with density, Calc. Var. Partial Differential Equations 31 (2008), no. 1, 27–46. MR 2342613 (2008m:49212) Zbl1126.49038
  68. [68] L. Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University Centre for Mathematical Analysis, Canberra, 1983. MR 756417 (87a:49001) Zbl0546.49019
  69. [69] P. Sternberg and K. Zumbrun, A Poincaré inequality with applications to volume-constrained area-minimizing surfaces, J. Reine Angew. Math. 503 (1998), 63–85. MR 1650327 (99g:58028) Zbl0967.53006
  70. [70] , On the connectivity of boundaries of sets minimizing perimeter subject to a volume constraint, Comm. Anal. Geom. 7 (1999), no. 1, 199–220. MR 1674097 (2000d:49062) Zbl0930.49024
  71. [71] V. N. Sudakov and B. S. Tirel’son, Extremal properties of half-spaces for spherically invariant measures, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 41 (1974), 14–24, 165, Problems in the theory of probability distributions, II. MR MR0365680 (51 #1932) 
  72. [72] J. Szarski, Differential inequalities, Monografie Matematyczne, Tom 43, Panstwowe Wydawnictwo Naukowe, Warsaw, 1965. MR 0190409 (32 #7822) 
  73. [73] M. Troyanov, Parabolicity of manifolds, Siberian Adv. Math. 9 (1999), no. 4, 125–150. MR 1749853 (2001e:31013) 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.