Electronic Only Electronic Research Announcements
Electronic Research Announcements
ISSN 1079-6762
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
 
Currently published by the American Institute of Mathematical Sciences as Electronic Research Announcements in Mathematical Sciences
 

Optimal regularity for quasilinear equations in stratified nilpotent Lie groups and applications

Author(s): Luca Capogna
Journal: Electron. Res. Announc. Amer. Math. Soc. 2 (1996), 60-68.
MSC (1991): Primary 35H05
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

Abstract: We announce the optimal $C^{1+\alpha }$ regularity of the gradient of weak solutions to a class of quasilinear degenerate elliptic equations in nilpotent stratified Lie groups of step two. As a consequence we also prove a Liouville type theorem for $1$-quasiconformal mappings between domains of the Heisenberg group $\mathbb {H}^{n}$.

References:

[CDG1]
L. Capogna, D. Danielli, and N. Garofalo, An embedding theorem and the Harnack inequality for nonlinear subelliptic equations, Comm. P.D.E. 18 (1993), 1765--1794. MR 94j:35038

[CDG2]
------, Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations, to appear in the Amer. J. of Math.

[D]
D. Danielli, Regularity at the boundary for solutions of nonlinear subelliptic equations, Indiana Univ. Math. Jour. 44 (1995), 269--286. CMP 95:14

[F]
G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Arkiv. für Math. 13 (1975), 161--207. MR 58:13215

[FS]
G. B. Folland and E. M. Stein, Estimates for the $\bar {\partial _{b}}$ complex and analysis on the Heisenberg group, Comm Pure and Appl. Math. 27 (1974), 459--522. MR 51:3719

[G1]
F. W. Gehring, Rings and quasiconformal mappings in space, Trans. of the American Mathematical Society 103 (1962), 353--393. MR 25:3166

[G2]
------, The $L^{p}$-integrability of the partial derivatives of a quasiconformal map, Acta Math. 130 (1973), 265--277. MR 53:5861

[Gi]
E. Giusti, Minimal surfaces and functions of bounded variation, vol. 80, Boston, Birkhäuser, Monographs in Math., 1984. MR 87a:58041

[K]
J. J. Kohn, Pseudo-differential operators and hypoellipticity, Proceedings of Symposia in Pure Mathematics XXIII (1973), 61--69. MR 49:3356

[KR1]
A. Korányi and H. M. Reimann, Quasiconformal mappings on the Heisenberg group, Inv. Math. 80 (1985), 309--338. MR 86m:32035

[KR2]
------, Horizontal normal vectors and conformal capacity of spherical rings in the Heisenberg group, Bull. Sci. Math. (2) 111 (1987), 3--21. MR 88e:22013

[KR3]
------, Foundations for the theory of quasiconformal mappings on the Heisenberg group, Adv. in Math. 111 (1995), 1--87. MR 96c:30021

[M]
C. B. Morrey, Jr., Multiple integrals in the calculus of variations, Springer-Verlag, New York, 1966. MR 34:2380

[Mo]
G. D. Mostow, Strong rigidity of locally symmetric spaces, Princeton University Press, Princeton, NJ, 1973. MR 52:5874

[P]
P. Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. Math. 129 (1989), 1--60. MR 90e:53058

[Pe]
J. Peetre, A theory of interpolation of normed spaces, Notas de Matématica 39 (1968). MR 39:4662

[R]
Y. G. Reshetnyak, Space mappings with bounded distortion, Transl. of Math. Mon., vol 73, American Mathematical Society, 1989. MR 90d:30067

[S]
E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, 1970. MR 44:7280

[T]
P. Tang, Quasiconformal homeomorphisms on CR 3-manifolds: regularity and extremality, to appear in Ann. Acad. Sci. Fenn. Ser. A. CMP 95:15

[X1]
C.-J. Xu, Subelliptic variational problems, Bull. Soc. Math. France 118 (1990), 147--169. MR 92b:49008

[X2]
------, Regularity for quasilinear second order subelliptic equations, Comm. Pure and Appl. Math. XLV (1992), 77--96. MR 93b:35042

Similar Articles:

Retrieve articles in Electronic Research Announcements with MSC (1991): 35H05

Retrieve articles in all Journals with MSC (1991): 35H05

Additional Information:

Luca Capogna
Affiliation: Department of Mathematics, Purdue University, West Lafayette, IN 47907
Email: capogna@math.purdue.edu

DOI: 10.1090/S1079-6762-96-00009-1
PII: S 1079-6762(96)00009-1
Received by editor(s): March 15, 1996
Additional Notes: Alfred P. Sloan Doctoral Dissertation Fellow.
Communicated by: Thomas Wolff
Copyright of article: Copyright 1996, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google