Abstract
Let M and N be compact smooth Riemannian manifolds without boundaries. Then, for a map u: M → N, we consider a class of energies which includes the popular Dirichlet energy and the more general p-energy. Geometric or physical questions motivate to investigate the critical points of such an energy or the corresponding heat flow problem. In the case of the Dirichlet energy, the heat flow problem has been intensively studied and is well understood by now. However, it has turned out that the case of the p-energy (p ≠ 2) is much more difficult in many respects. We give a survey of the known results for the p-harmonic flow and indicate how these results can be extended to a larger class of energy types by using Young measure techniques which have recently been developed for quasilinear problems.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J.M. Ball: A version of the fundamental theorem for Young measures. In: Partial differential equations and continuum models of phase transitions: Proceedings of an NSF-CNRS joint seminar. Springer, 1989
F. Bethuel: The approximation problem for Sobolev maps between manifolds. Acta Math. 167, 153–206 (1991)
F. Bethuel, X. Zheng: Density of smooth functions between two manifolds in Sobolev spaces. J. Funct. Anal. 80, 60–75 (1988)
K.-C. Chang, W.-Y. Ding, R. Ye, R.: Finite-time blow-up of the heat flow of harmonic maps from surfaces. J. Differ. Geom. 36, No. 2, 507–515 (1992)
Y. Chen: The weak solutions to the evolution problem of harmonic maps. Math. Z. 201, 69–74 (1989)
Y. Chen, M.-C. Hong, N. Hungerbühler: Heat flow of p-harmonic maps with values into spheres. Math. Z. 215, 25–35 (1994)
P. Courilleau, F. Demengel: Heat flow for p-harmonic maps with values in the circle. Nonlinear Anal. 41, no. 5–6, Ser. A: Theory Methods, 689–700 (2000)
G. Dolzmann, N. Hungerbühler, S. Müller: Non-linear elliptic systems with measure-valued right-hand side. Math. Z. 226, 545–574 (1997)
F. Duzaar, M. Fuchs: On removable singularities of p-harmonic maps. Analyse non linéaire, 7, No. 5 385–405 (1990)
F. Duzaar, M. Fuchs: Existence and regularity of functions which minimize certain energies in homotopy classes of mappings. Asymptotic Analysis 5, 129–144 (1991)
J. Eells, L. Lemaire: Another report on harmonic maps. Bull. London Math. Soc. 20, 385–524 (1988)
J. Eells, J.H. Sampson: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–169 (1964)
J. Eells, J.C. Wood: Restrictions on harmonic maps of surfaces. Topology 15, 263–266 (1976)
A. Fardoun, R. Regbaoui: Équation de la chaleur pour les applications p-harmoniques entre variétés riemanniennes compactes. (French) C. R. Acad. Sci. Paris Sér. I Math. 333, no. 11, 979–984 (2001)
A. Fardoun, R. Regbaoui: Heat flow for p-harmonic map with small initial data. To appear in Calc. Var. Partial Differential Equations
A. Fardoun, R. Regbaoui: Heat flow for p-harmonic maps between compact Riemannian manifolds. To appear in Indiana Univ. Math. J.
H. Federer: Geometric measure theory. Springer, 1969
M. Fuchs: Some regularity theorems for mappings which are stationary points of the p-energy functional. Analysis 9, 127–143 (1989)
M. Fuchs: p,-harmonic obstacle problems. Part III: Boundary regularity. Annali Mat. Pura Applicata 156, 159–180 (1990)
R.S. Hamilton: Harmonic maps of manifolds with boundary. Lect. Notes Math. 471, Springer, Berlin, 1975
R. Hardt, F.H. Lin: Mappings minimizing the LP, norm of the gradient. Comm. Pure and appl. Math 15, 555–588 (1987)
N. Hungerbühler: p,-harmonic flow. Diss. Math. Wiss. ETH Zürich, Nr. 10740, 1994. Ref.: Michael Struwe; Korref.: Jürgen Moser, Zürich, 1994
N. Hungerbühler: Global weak solutions of the p-harmonic flow into homogeneous spaces. Indiana Univ. Math. J. 45/1, 275–288 (1996)
N. Hungerbühler: Compactness properties of the p-harmonic flow into homogeneous spaces. Nonlinear Anal. 28/5, 793–798 (1997)
N. Hungerbühler: Non-uniqueness for the p-harmonic flow. Canad. Math. Bull. 40/2, 174–182 (1997)
N. Hungerbühler: m-harmonic flow. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) XXIV, 4, 593–631 (1997)
N. Hungerbühler: A refinement of Ball’s Theorem on Young measures. New York J. Math. 3, 48–53 (1997)
N. Hungerbühler: Young measures and nonlinear PDEs. Habilitationsschrift ETH Zürich, 2000
N. Hungerbühler: Quasilinear parabolic systems in divergence form with weak mono-tonicity. Duke Math. J. 107/3, 497–520 (2001)
L. Lemaire: Applications harmoniques de surfaces riemannienne. J. Diff. Geom. 13, 51–78 (1978)
X.G. Liu, S.H. Li: The p-harmonic heat flow with potential into homogeneous spaces. Acta Math. Sin. (Engl. Ser.) 18, no. 1, 21–26 (2002)
X.G. Liu: A note on heat flow of p-harmonic mappings. (Chinese) Kexue Tongbao (Chinese) 42, no. 1, 15–18 (1997)
X.G. Liu: A remark on p-harmonic heat flows. Chinese Sci. Bull. 42, no. 6, 441–444 (1997)
M. Misawa: On regularity for heat flows for p-harmonic maps. Proceedings of the Korea-Japan Partial Differential Equations Conference (Taejon, 1996), 14 pp., Lecture Notes Ser., 39, Seoul Nat. Univ., Seoul, 1997
M. Misawa: Existence and regularity results for the gradient flow for p-harmonic maps. Electron. J. Differential Equations, No. 36, 17 pp. (1998)
M. Misawa: Existence and regularity results for the gradient flow for p-harmonic maps. Regularity, blowup and related properties of solutions to nonlinear evolution equations (Japanese). (Kyoto, 1997). Surikaisekikenkyüsho Kókyüroku No. 1045, 5772 (1998)
M. Misawa: On the p-harmonic flow into spheres in the singular case. Nonlinear Anal. 50, no. 4, Ser. A: Theory Methods, 485–494 (2002)
J. Sacks, K. Uhlenbeck: The existence of minimal immersions of 2-spheres. Ann. of Math. 113, 1–24 (1981)
R.S. Schoen, K. Uhlenbeck: Approximation theorems for Sobolev mappings, preprint
P. Strzelecki: Regularity of p-harmonic maps from the p-dimensional ball into a sphere. Manuscr. Math. 82, No.3–4, 407–415 (1994)
P. Strzelecki: Stationary p-harmonic maps into spheres. Janeczko, Stanislaw (ed.) et al., Singularities and differential equations. Proceedings of a symposium, Warsaw, Poland. Warsaw: Polish Academy of Sciences, Inst. of Mathematics, Banach Cent. Publ. 33, 383–393 (1996)
E. Zeidler: Nonlinear functional analysis and its applications,Vol. II/B: Nonlinear Monotone Operators. New York: Springer, 1990
C.Q. Zhou: Compactness properties of heat flows for weakly p-harmonic maps. (Chinese) Acta Math. Sinica 41, no. 2, 327–336 (1998)
C.Q. Zhou, Y.D. Wang: Existence and nonuniqueness for the flow of p-harmonic maps. (Chinese) Acta Math. Sinica 41, no. 3, 511–516 (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer Basel AG
About this paper
Cite this paper
Hungerbühler, N. (2004). Heat Flow into Spheres for a Class of Energies. In: Baird, P., Fardoun, A., Regbaoui, R., El Soufi, A. (eds) Variational Problems in Riemannian Geometry. Progress in Nonlinear Differential Equations and Their Applications, vol 59. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7968-2_4
Download citation
DOI: https://doi.org/10.1007/978-3-0348-7968-2_4
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9640-5
Online ISBN: 978-3-0348-7968-2
eBook Packages: Springer Book Archive