New properties of convex functions in the Heisenberg group
HTML articles powered by AMS MathViewer
- by Nicola Garofalo and Federico Tournier PDF
- Trans. Amer. Math. Soc. 358 (2006), 2011-2055 Request permission
Abstract:
We prove some new properties of the weakly $H$-convex functions recently introduced by Danielli, Garofalo and Nhieu. As an interesting application of our results we prove a theorem of Busemann-Feller-Alexandrov type in the Heisenberg groups $\mathbb {H}^n$, $n=1,2$.References
- A. D. Aleksandrov, Certain estimates for the Dirichlet problem, Soviet Math. Dokl. 1 (1961), 1151–1154. MR 0147776
- A. D. Aleksandrov, Impossibility of general estimates of solutions and of uniqueness conditions for linear equations with norms weaker than those of $L_{n}$, Vestnik Leningrad. Univ. 21 (1966), no. 13, 5–10 (Russian, with English summary). MR 0197952
- Luigi Ambrosio and Valentino Magnani, Weak differentiability of BV functions on stratified groups, Math. Z. 245 (2003), no. 1, 123–153. MR 2023957, DOI 10.1007/s00209-003-0530-2
- Gunnar Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat. 6 (1967), 551–561 (1967). MR 217665, DOI 10.1007/BF02591928
- Gunnar Aronsson, On the partial differential equation $u_{x}{}^{2}\!u_{xx} +2u_{x}u_{y}u_{xy}+u_{y}{}^{2}\!u_{yy}=0$, Ark. Mat. 7 (1968), 395–425 (1968). MR 237962, DOI 10.1007/BF02590989
- I. Ja. Bakel′man, On the theory of quasilinear elliptic equations, Sibirsk. Mat. Ž. 2 (1961), 179–186 (Russian). MR 0126604
- I. Ja. Bakel′man, Geometricheskie metody resheniya èllipticheskikh uravneniĭ, Izdat. “Nauka”, Moscow, 1965 (Russian). MR 0194727
- Ilya J. Bakelman, Convex analysis and nonlinear geometric elliptic equations, Springer-Verlag, Berlin, 1994. With an obituary for the author by William Rundell; Edited by Steven D. Taliaferro. MR 1305147, DOI 10.1007/978-3-642-69881-1
- Zoltán M. Balogh and Matthieu Rickly, Regularity of convex functions on Heisenberg groups, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), no. 4, 847–868. MR 2040646
- André Bellaïche and Jean-Jacques Risler (eds.), Sub-Riemannian geometry, Progress in Mathematics, vol. 144, Birkhäuser Verlag, Basel, 1996. MR 1421821, DOI 10.1007/978-3-0348-9210-0
- Thomas Bieske, On $\infty$-harmonic functions on the Heisenberg group, Comm. Partial Differential Equations 27 (2002), no. 3-4, 727–761. MR 1900561, DOI 10.1081/PDE-120002872
- Thomas Bieske and Luca Capogna, The Aronsson-Euler equation for absolutely minimizing Lipschitz extensions with respect to Carnot-Carathéodory metrics, Trans. Amer. Math. Soc. 357 (2005), no. 2, 795–823. MR 2095631, DOI 10.1090/S0002-9947-04-03601-3
- Jean-Michel Bony, Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier (Grenoble) 19 (1969), no. fasc. 1, 277–304 xii (French, with English summary). MR 262881
- Luis A. Caffarelli and Xavier Cabré, Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. MR 1351007, DOI 10.1090/coll/043
- Luca Capogna, Donatella Danielli, and Nicola Garofalo, The geometric Sobolev embedding for vector fields and the isoperimetric inequality, Comm. Anal. Geom. 2 (1994), no. 2, 203–215. MR 1312686, DOI 10.4310/CAG.1994.v2.n2.a2
- Lawrence J. Corwin and Frederick P. Greenleaf, Representations of nilpotent Lie groups and their applications. Part I, Cambridge Studies in Advanced Mathematics, vol. 18, Cambridge University Press, Cambridge, 1990. Basic theory and examples. MR 1070979
- Michael Cowling, Anthony H. Dooley, Adam Korányi, and Fulvio Ricci, $H$-type groups and Iwasawa decompositions, Adv. Math. 87 (1991), no. 1, 1–41. MR 1102963, DOI 10.1016/0001-8708(91)90060-K
- Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67. MR 1118699, DOI 10.1090/S0273-0979-1992-00266-5
- Jacek Cygan, Subadditivity of homogeneous norms on certain nilpotent Lie groups, Proc. Amer. Math. Soc. 83 (1981), no. 1, 69–70. MR 619983, DOI 10.1090/S0002-9939-1981-0619983-8
- Bernard Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences, vol. 78, Springer-Verlag, Berlin, 1989. MR 990890, DOI 10.1007/978-3-642-51440-1
- Donatella Danielli, Nicola Garofalo, and Duy-Minh Nhieu, Notions of convexity in Carnot groups, Comm. Anal. Geom. 11 (2003), no. 2, 263–341. MR 2014879, DOI 10.4310/CAG.2003.v11.n2.a5
- Donatella Danielli, Nicola Garofalo, and Duy-Minh Nhieu, On the best possible character of the $L^Q$ norm in some a priori estimates for non-divergence form equations in Carnot groups, Proc. Amer. Math. Soc. 131 (2003), no. 11, 3487–3498. MR 1991760, DOI 10.1090/S0002-9939-03-07105-3
- —, Minimal surfaces in Carnot groups, preprint, 2003.
- D. Danielli, N. Garofalo, D. M. Nhieu, and F. Tournier, The theorem of Busemann-Feller-Alexandrov in Carnot groups, Comm. Anal. Geom. 12 (2004), no. 4, 853–886. MR 2104079
- Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
- Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
- G. B. Folland, A fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc. 79 (1973), 373–376. MR 315267, DOI 10.1090/S0002-9904-1973-13171-4
- G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), no. 2, 161–207. MR 494315, DOI 10.1007/BF02386204
- G. B. Folland and E. M. Stein, Estimates for the $\bar \partial _{b}$ complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429–522. MR 367477, DOI 10.1002/cpa.3160270403
- G. B. Folland and Elias M. Stein, Hardy spaces on homogeneous groups, Mathematical Notes, vol. 28, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. MR 657581
- N. Garofalo, Analysis and Geometry of Carnot-Carathéodory Spaces, With Applications to Pde’s, Birkhäuser, book in preparation.
- Nicola Garofalo and Duy-Minh Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math. 49 (1996), no. 10, 1081–1144. MR 1404326, DOI 10.1002/(SICI)1097-0312(199610)49:10<1081::AID-CPA3>3.0.CO;2-A
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
- Cristian E. Gutiérrez, The Monge-Ampère equation, Progress in Nonlinear Differential Equations and their Applications, vol. 44, Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1829162, DOI 10.1007/978-1-4612-0195-3
- C. E. Gutierrez & A. Montanari, Maximum and comparison principles for convex functions on the Heisenberg group, preprint, first posted on May 31, 2003, to the webpage www.dm.unibo.it/$\sim$montanar
- Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 222474, DOI 10.1007/BF02392081
- Alston S. Householder, The theory of matrices in numerical analysis, Blaisdell Publishing Co. [Ginn and Co.], New York-Toronto-London, 1964. MR 0175290
- Robert Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Rational Mech. Anal. 123 (1993), no. 1, 51–74. MR 1218686, DOI 10.1007/BF00386368
- Aroldo Kaplan, Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. Amer. Math. Soc. 258 (1980), no. 1, 147–153. MR 554324, DOI 10.1090/S0002-9947-1980-0554324-X
- N. V. Krylov, Sequences of convex functions, and estimates of the maximum of the solution of a parabolic equation, Sibirsk. Mat. Ž. 17 (1976), no. 2, 290–303, 478 (Russian). MR 0420016
- Guozhen Lu, Juan J. Manfredi, and Bianca Stroffolini, Convex functions on the Heisenberg group, Calc. Var. Partial Differential Equations 19 (2004), no. 1, 1–22. MR 2027845, DOI 10.1007/s00526-003-0190-4
- V. Magnani, Lipschitz continuity, Alexandrov theorem, and characterizations for $H$-convex functions, preprint, September 2003.
- Richard Montgomery, A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, vol. 91, American Mathematical Society, Providence, RI, 2002. MR 1867362, DOI 10.1090/surv/091
- Pierre Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129 (1989), no. 1, 1–60 (French, with English summary). MR 979599, DOI 10.2307/1971484
- Carlo Pucci, Operatori ellittici estremanti, Ann. Mat. Pura Appl. (4) 72 (1966), 141–170 (Italian, with English summary). MR 208150, DOI 10.1007/BF02414332
- Carlo Pucci, Limitazioni per soluzioni di equazioni ellittiche, Ann. Mat. Pura Appl. (4) 74 (1966), 15–30 (Italian, with English summary). MR 214905, DOI 10.1007/BF02416445
- Elias 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. MR 1232192
- Srdjan Stojanovic, Computational financial mathematics using Mathematica$^\circledR$, Birkhäuser Boston, Inc., Boston, MA; TELOS. The Electronic Library of Science, Santa Clara, CA, 2003. Optimal trading in stocks and options; With 1 CD-ROM (Windows and Macintosh). MR 1930223, DOI 10.1007/978-1-4612-0043-7
- —, Optimal momentum edging via hypoelliptic reduced Monge-Ampère PDE’s, preprint, 2003.
- Neil S. Trudinger and Xu-Jia Wang, Hessian measures. I, Topol. Methods Nonlinear Anal. 10 (1997), no. 2, 225–239. Dedicated to Olga Ladyzhenskaya. MR 1634570, DOI 10.12775/TMNA.1997.030
- V. S. Varadarajan, Lie groups, Lie algebras, and their representations, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. MR 0376938
- C. Wang, The Aronsson equation for absolute minimizers of $L^\infty$ functionals associated with vector fields satisfying Hörmander’s condition, preprint, May 2003.
- —, The comparison principle for viscosity solutions of fully nonlinear sub-elliptic equations in Carnot groups, preprint, July 2003.
- Changyou Wang, Viscosity convex functions on Carnot groups, Proc. Amer. Math. Soc. 133 (2005), no. 4, 1247–1253. MR 2117228, DOI 10.1090/S0002-9939-04-07836-0
Additional Information
- Nicola Garofalo
- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
- MR Author ID: 71535
- Email: garofalo@math.purdue.edu
- Federico Tournier
- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
- Email: tournier@math.purdue.edu, fedeleti@aim.com
- Received by editor(s): February 13, 2004
- Published electronically: December 20, 2005
- Additional Notes: The first author was supported in part by NSF Grants No. DMS-0070492 and No. DMS-0300477
This work was presented by the first author in a Colloquium lecture at the University of Missouri, Columbia, in April 2003, and at the AMS Meeting at San Francisco State University in May 2003 - © Copyright 2005 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 358 (2006), 2011-2055
- MSC (2000): Primary 35H20; Secondary 26B25, 20F18
- DOI: https://doi.org/10.1090/S0002-9947-05-04016-X
- MathSciNet review: 2197446