Uniform Gaussian Bounds for Subelliptic Heat Kernels and an Application to the Total Variation Flow of Graphs over Carnot Groups

Luca Capogna ; Giovanna Citti ; Maria Manfredini

Analysis and Geometry in Metric Spaces, Tome 1 (2013), p. 255-275 / Harvested from The Polish Digital Mathematics Library

Résumé

In this paper we study heat kernels associated with a Carnot group G, endowed with a family of collapsing left-invariant Riemannian metrics σε which converge in the Gromov- Hausdorff sense to a sub-Riemannian structure on G as ε→ 0. The main new contribution are Gaussian-type bounds on the heat kernel for the σε metrics which are stable as ε→0 and extend the previous time-independent estimates in [16]. As an application we study well posedness of the total variation flow of graph surfaces over a bounded domain in a step two Carnot group (G; σε ). We establish interior and boundary gradient estimates, and develop a Schauder theory which are stable as ε → 0. As a consequence we obtain long time existence of smooth solutions of the sub-Riemannian flow (ε = 0), which in turn yield sub-Riemannian minimal surfaces as t → ∞.

Détail

Publié le : 2013-01-01

Zbl 1275.53055

EUDML-ID : urn:eudml:doc:266636

Bibliographie

[1] F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, The Dirichlet problem for the total variation flow. J. Funct. Anal. 180 (2001), no. 2, 347-403. | Zbl 0973.35109

[2] F. Andreu, V. Caselles and J. M. Mazón Ruiz, Parabolic quasilinear equations minimizing linear growth functionals, Progr. Math., 223, Birkhauser, Basel, 2004 | Zbl 1053.35002

[3] D. G. Aronson. and J. Serrin, Local behavior of solutions of quasilinear parabolic equations, Arch. Rational Mech. Anal., 25 (1967), 81-122. | Zbl 0154.12001

[4] D. Barbieri. G. Citti, G. Cocci and A. Sarti, A cortical-inspired geometry for contour perception and motion integration, submitted paper. | Zbl 1291.92074

[5] G. Bellettini, V. Caselles and M. Novaga, The total variation flow in RN, J. Differential Equations 184 (2002), no. 2, 475-525. | Zbl 1036.35099

[6] A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Uniform Gaussian estimates of the fundamental solutions for heat operators on Carnot groups, Adv. Differential Equations, 7 (2002), 1153-1192. | Zbl 1036.35061

[7] A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Fundamental solutions for non-divergence form operators on stratified groups, Trans. Amer. Math. Soc. 356 (2004), no. 7, 2709-2737. | Zbl 1072.35011

[8] L. Capogna and G. Citti, Generalized mean curvature flow in Carnot groups, Comm. Partial Differential Equations 34 (2009), no. 7-9, 937-956. | Zbl 1179.35097

[9] L. Capogna, G. Citti and M. Manfredini, Regularity of non-characteristic minimal graphs in the Heisenberg group H1, Indiana Univ. Math. J. 58 (5) (2009) 2115-2160. [WoS] | Zbl 1182.35087

[10] L. Capogna, G. Citti and M. Manfredini, Smoothness of Lipschitz intrinsic minimal graphs in the Heisenberg group Hn, n > 1 J. Reine Angew. Math. (Crelle’s Journal) (2010), 648, 75-110. | Zbl 1217.53060

[11] L. Capogna, G. Citti and G. Rea, A sub-elliptic analogue of Aronson-Serrin Harnack inequality, to appear in Mathematische Annalen, (2013). | Zbl 1282.35204

[12] L. Capogna, G. Citti and C. Senni Guidotti Magnani,Sub-Riemannian Heat Kernels and Mean Curvature of graphs, Jour. of Functional Analysis, 264 (2013), no. 8 1899-1928. | Zbl 1277.53062

[13] L. Capogna, D. Danielli, J. Tyson and S. Pauls, An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, Birkhauser, 2007. | Zbl 1138.53003

[14] J.-H. Cheng, J.-F. Hwang and P. Yang, Regularity of C1 smooth surfaces with prescribed p-mean curvature in the Heisenberg group, Math. Ann. 344 (2009), no. 1, 1-35. [WoS] | Zbl 1171.35026

[15] J.H. Cheng and J.F. Hwang, A. Malchiodi and P. Yang, Minimal surfaces in pseudohermitian geometry, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005), 129-177. | Zbl 1158.53306

[16] G. Citti and M. Manfredini, Uniform estimates of the fundamental solution for a family of hypoelliptic operators, Potential Anal. 25 (2006), no. 2, 147-164. | Zbl 1112.35047

[17] G. Citti, A. Pascucci and S. Polidoro, On the regularity of solutions to a nonlinear ultraparabolic equation arising in mathematical finance. Differential Integral Equations 14 (2001), no. 6, 701-738. | Zbl 1013.35046

[18] G. Citti and A. Sarti, A cortical based model of perceptual completion in the Roto-Translation space, Journal of Mathematical Analysis and Vision, (2006), 24, 307 - 326. | Zbl 1088.92008

[19] T. Chan, S. Esedoglu, F. Park and A. Yip, Total Variation Image Restoration: Overview and Recent Developments Handbook of Mathematical Models in Computer Vision, (2006), 17-31.

[20] R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homognes. (french) Etude de certaines intégrales singulières, Lecture Notes in Mathematics, Vol. 242, Springer-Verlag, 1971. | Zbl 0224.43006

[21] D. Danielli, N. Garofalo, D.M. Nhieu and S.D. Pauls, Instability of graphical strips and a positive answer to the Bernstein problem in the Heisenberg group H1, J. Differ. Geom, (2009), 81(2), 251-295. | Zbl 1161.53024

[22] N. Dirr, F. Dragoni and M. von Renesse, Evolution by mean curvature flow in sub-Riemannian geometries: A stochastic approach, Communications on Pure and Applied Mathematics, 9 (2), (2010) pp. 307-326. | Zbl 1202.53064

[23] R. Duits and E. M. Franken, Left invariant parabolic evolution equations on SE(2) and contour enhancement via invertible orientation scores, part I: Linear left-invariant diffusion equations on SE(2), Quarterly of Applied Mathematics, AMS, (2010), 68, 255-292. | Zbl 1202.35334

[24] R. Duits and E. Franken, Left invariant parabolic evolution equations on SE(2) and contour enhancement via invertible orientation scores, part II: Nonlinear left-invariant diffusion equations on invertible orientation scores,Quarterly of Applied Mathematics, AMS, (2010), 68, 293-331. | Zbl 1205.35326

[25] R. Duits, H. Fuhr, B.J. Janssen, M. Bruurmijn, L.M.J. Florack and H.A.C.van Assen, Evolution Equations on Gabor Transforms and their Applications, Applied and Computational Harmonic Analysis, accepted for publication, to appear | Zbl 1296.35204

[26] G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 2 (1975), no. 13, 161-207. | Zbl 0312.35026

[27] G. B. Folland, Harmonic Analysis in Phase Space, Annals of Mathematics Studies, 122, Princeton Univ. Press, Princeton, 1989. | Zbl 0682.43001

[28] G. B. Folland and E. M. Stein, Hardy spaces on homogeneous groups, Princeton University Press, Princeton, New Jersey, 1982. | Zbl 0508.42025

[29] A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J. 1964 xiv+347 pp. | Zbl 0144.34903

[30] M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Birkhauser Boston Inc., Boston, MA, 1999, Edited by J. LaFontaine and P. Pansu. Based on the 1981. French original, Translated from the French by Sean Michael Bates. | Zbl 0953.53002

[31] A. A. Grigor’yan, The heat equation on noncompact Riemannian manifolds. (Russian) Mat. Sb. 182 (1991), no. 1, 55-87; translation in Math. USSR-Sb. 72 (1992), no. 1, 47-77. | Zbl 0743.58031

[32] L. Hormander, Hypoelliptic second order differential equations, Acta Math. (1967), no. 119, 147-171. | Zbl 0156.10701

[33] D. S. Jerison and A. Sanchez-Calle, Estimates for the heat kernel for a sum of squares of vector fields, Indiana Univ. Math. J. 35 (1986), no. 4, 835-854. | Zbl 0639.58026

[34] O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Uraltseva, Linear and quasilinear equations of parabolic type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 (Russian).

[35] G. M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co. Inc., River Edge, NJ, 1996. | Zbl 0884.35001

[36] F. Ferrari, J. Manfredi and Q. Liu, On the horizontal Mean Curvature Flow for Axisymmetric surfaces in the Heisenberg Group, preprint (2012). | Zbl 1304.35389

[37] J. S. Moll, The anisotropic total variation flow. (English summary) Math. Ann. 332 (2005), no. 1, 177-218. | Zbl 1109.35061

[38] R. Monti and M. Rickly, Convex isoperimetric sets in the Heisenberg group, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8 (2009), no. 2, 391-415. | Zbl 1170.49037

[39] J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101-134. | Zbl 0149.06902

[40] A. Nagel, E. M. Stein, and S. Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985), no. 1-2, 103-147. | Zbl 0578.32044

[41] S.D. Pauls: Minimal surfaces in the Heisenberg group, Geom. Dedic., (2004), 104, 201-231.

[42] M. Ritoré and C. Rosales, Area-stationary surfaces in the Heisenberg group H1, Adv. Math. 219 (2008), 633-671.

[43] L. P. Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), no. 3-4, 247-320. | Zbl 0346.35030

[44] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D 60 (1992), 259-268. | Zbl 0780.49028

[45] L. Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequalities, Internat. Math. Res. Notices 1992 (1992), no. 2, 27-38. | Zbl 0769.58054

[46] F. Serra Cassano and D. Vittone, Graphs of bounded variation, existence and local boundedness of non-parametric minimal surfaces in Heisenberg groups, (accepted for publication in Advances in Calculus of Variations). | Zbl 1304.49084

[47] E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. | Zbl 0821.42001