Proceedings of the American Mathematical Society

Author(s): Luca Capogna; Jalal Shatah
Journal: Proc. Amer. Math. Soc. 136 (2008), 1619-1629.
MSC (2000): Primary 35L55, 53C17
Posted: January 28, 2008
Retrieve article in: PDF DVI PostScript

Abstract: We introduce a notion of wave maps with a target in the sub-Riemannian Heisenberg group and study their relation with Riemannian wave maps with range in Lagrangian submanifolds. As an application we establish existence and eventually ill-posedness of the corresponding Cauchy problem.

1.: B. Aebischer, M. Borer, M. Kälin, Ch. Leuenberger and H. M. Reimann, Symplectic Geometry, Progr. Math., 124 (1994), Birkhäuser. MR 1296462 (96a:58082)
2.: L. Ambrosio, Transport equation and Cauchy problem for vector fields. Invent. Math. 158 (2004), no. 2, 227-260. MR 2096794 (2005f:35127)
3.: H. Bahouri and J.-Y. Chemin, Équations de transport relatives á des champs de vecteurs non-lipschitziens et mécanique des fluides (French) [Transport equations for non-Lipschitz vector fields and fluid mechanics]. Arch. Rational Mech. Anal. 127 (1994), no. 2, 159-181. MR 1288809 (95g:35164)
4.: L. Capogna, D. Danielli, S. Pauls and J. Tyson, An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem, Progr. Math., 259, Birkhäuser, Basel, 2007. MR 2312336
5.: L. Capogna and F.-H. Lin, Legendrian energy minimizers. I: Heisenberg group target, Calc. Var. Partial Differential Equations 12 (2001), no. 2, 145-171. MR 1821235 (2002k:58032)
6.: G. Citti and A. Sarti, A cortical based model of perceptual completion in the roto-translation space. J. Math. Imaging Vision 24 (2006), no. 3, 307-326. MR 2235475 (2007b:92016)
7.: R. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989), no. 3, 511-547. MR 1022305 (90j:34004)
8.: G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Princeton Univ. Press, Princeton; University of Tokyo Press, Tokyo, 1982. MR 657581 (84h:43027)
9.: R. T. Glassey, The Cauchy problem in kinetic theory, SIAM, Philadelphia, 1996. MR 1379589 (97i:82070)
10.: R. T. Glassey and W. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Rat. Mech. Anal. 92 (1986), 56-90. MR 816621 (87j:82064)
11.: M. Gromov, Carnot-Caratheodory spaces seen from within, in Sub-Riemannian geometry, 79-323, Progr. Math., 144, Birkhäuser, Basel, 1996. MR 1421823 (2000f:53034)
12.: A. Koranyi, Geometric aspects of analysis on the Heisenberg group. Topics in modern harmonic analysis, Vols. I, II (Turin/Milan, 1982), 209-258, Ist. Naz. Alta Mat. Francesco Severi, Rome, 1983. MR 748865 (85h:32055)
13.: N. Korevaar and R. Schoen, Sobolev spaces and harmonic maps for metric space targets. Comm. Anal. Geom. 1 (1993), no. 3-4, 561-659. MR 1266480 (95b:58043)
14.: J. Petitot, The neurogeometry of pinwheels as a sub-Riemannian contact structure, J. Physiology 97 (2003), 265-309.
15.: J. Shatah, Weak solutions and development of singularities of the SU(2) $\sigma$ -model, Comm. Pure and Appl. Math. 41 (1988), 459-469. MR 933231 (89f:58044)
16.: J. Shatah and M. Struwe, Geometric wave equations, CIMS Lecture Notes, 2, Courant Institute of Mathematical Sciences, New York; Amer. Math. Soc., Providence, RI, 1998. MR 1674843 (2000i:35135)
17.: J. Shatah and C. Zeng, Constrained wave equations and wave maps, Comm. in Math. Phys. 239 (2003), 383-404. MR 2000924 (2004g:58037)
18.: H. Sussmann, Geometry and optimal control. Mathematical control theory, 140-198, Springer, New York, 1999. MR 1661472 (99k:49051)

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35L55, 53C17

Luca Capogna
Affiliation: Department of Mathematics, University of Arkansas, Fayetteville, Arkansas 72701
Email: lcapogna@comp.uark.edu

Jalal Shatah
Affiliation: Courant Institute of Mathematical Sciences, 251 Mercer St., New York, New York 10012
Email: shatah@cims.nyu.edu

DOI: 10.1090/S0002-9939-08-09302-7
PII: S 0002-9939(08)09302-7
Keywords: Wave maps, Heisenberg group
Received by editor(s): September 19, 2006
Posted: January 28, 2008
Additional Notes: The first author was partially supported by a National Science Foundation CAREER grant and by an Arkansas Science and Technology Authority grant.
The second author was partially supported by the National Science Foundation grant DMS 0203485.
Communicated by: David S. Tartakoff
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS