Optimal regularity for quasilinear equations in stratified nilpotent Lie groups and applications
Author:
Luca Capogna
Journal:
Electron. Res. Announc. Amer. Math. Soc. 2 (1996), 60-68
MSC (1991):
Primary 35H05
DOI:
https://doi.org/10.1090/S1079-6762-96-00009-1
MathSciNet review:
1405970
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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 $\mathbb {H}^{n}$.
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- ---, Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations, to appear in the Amer. J. of Math.
- D. Danielli, Regularity at the boundary for solutions of nonlinear subelliptic equations, Indiana Univ. Math. Jour. 44 (1995), 269–286.
- G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Arkiv. für Math. 13 (1975), 161–207.
- G. B. Folland and E. M. Stein, Estimates for the $\bar {\bar {\partial }_b}$ complex and analysis on the Heisenberg group, Comm Pure and Appl. Math. 27 (1974), 459–522.
- F. W. Gehring, Rings and quasiconformal mappings in space, Trans. of the American Mathematical Society 103 (1962), 353–393.
- ---, The $L^{p}$-integrability of the partial derivatives of a quasiconformal map, Acta Math. 130 (1973), 265–277.
- E. Giusti, Minimal surfaces and functions of bounded variation, vol. 80, Boston, Birkhäuser, Monographs in Math., 1984.
- J. J. Kohn, Pseudo-differential operators and hypoellipticity, Proceedings of Symposia in Pure Mathematics XXIII (1973), 61–69.
- A. Korányi and H. M. Reimann, Quasiconformal mappings on the Heisenberg group, Inv. Math. 80 (1985), 309–338.
- ---, Horizontal normal vectors and conformal capacity of spherical rings in the Heisenberg group, Bull. Sci. Math. (2) 111 (1987), 3–21.
- ---, Foundations for the theory of quasiconformal mappings on the Heisenberg group, Adv. in Math. 111 (1995), 1–87.
- C. B. Morrey, Jr., Multiple integrals in the calculus of variations, Springer-Verlag, New York, 1966.
- G. D. Mostow, Strong rigidity of locally symmetric spaces, Princeton University Press, Princeton, NJ, 1973.
- P. Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. Math. 129 (1989), 1–60.
- J. Peetre, A theory of interpolation of normed spaces, Notas de Matématica 39 (1968).
- Y. G. Reshetnyak, Space mappings with bounded distortion, Transl. of Math. Mon., vol 73, American Mathematical Society, 1989.
- E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, 1970.
- P. Tang, Quasiconformal homeomorphisms on CR 3-manifolds: regularity and extremality, to appear in Ann. Acad. Sci. Fenn. Ser. A.
- C.-J. Xu, Subelliptic variational problems, Bull. Soc. Math. France 118 (1990), 147–169.
- ---, Regularity for quasilinear second order subelliptic equations, Comm. Pure and Appl. Math. XLV (1992), 77–96.
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Additional Information
Luca Capogna
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907
MR Author ID:
336615
Email:
capogna@math.purdue.edu
Received by editor(s):
March 15, 1996
Additional Notes:
Alfred P. Sloan Doctoral Dissertation Fellow.
Communicated by:
Thomas Wolff
Article copyright:
© Copyright 1996
American Mathematical Society
