A remark on generalized harmonic maps into spheres. (English) Zbl 0932.35071
Two known theorems on generalized harmonic maps are proved in a new direct way using a improved Wente’s inequality, which is also proved in this paper. With help of this inequality the regularity of a generalized harmonic map on spheres is shown, a uniqueness theorem for the harmonic map is proved and finally an \(\varepsilon\)-condition for the harmonic map is found. Some of the theorems are also generalized on compact manifolds.
Reviewer: Thoralf Chrobok (Berlin)
References:
| [1] | Almeida, L., The regularity problem for generalized harmonic maps into homogenous spaces, Calc. Var., 3, 193-242 (1995) · Zbl 0820.58013 |
| [2] | T. Aubin, Nonlinear analysis on manifolds, Monge-Ampère Equations, Grundlehren, vol. 252, Springer, Berlin, 1982.; T. Aubin, Nonlinear analysis on manifolds, Monge-Ampère Equations, Grundlehren, vol. 252, Springer, Berlin, 1982. · Zbl 0512.53044 |
| [3] | Bethuel, F., Un résultat de régularité pour solutions de l’équation des surfacesà courbure moyenne préscrite, C. R. Acad. Sci. Paris, 314, 1003-1007 (1992) · Zbl 0795.35024 |
| [4] | Bethuel, F.; Ghidaglia, J. M., Improved regularity of elliptic equations involving jacobians and applications, J. Math. Pure. Appl., 72, 441-475 (1993) · Zbl 0831.35025 |
| [5] | H. Brezis et J.M. Coron, Multiple solutions of \(H\); H. Brezis et J.M. Coron, Multiple solutions of \(H\) · Zbl 0537.49022 |
| [6] | R. Coifman, P.L. Lions, Y. Meyer et S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pure Appl. 72 (1993) 247-286.; R. Coifman, P.L. Lions, Y. Meyer et S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pure Appl. 72 (1993) 247-286. · Zbl 0864.42009 |
| [7] | Hélein, F., Régularité des applications faiblement harmoniques entre une surface et une sphère, C. R. Acad. Sci. Paris, 311, 519-524 (1990) · Zbl 0728.35014 |
| [8] | F. Hélein, Applications harmoniques, lois des conservation et repère mobile, Diderotéditeur, Paris- New York-Amsterdam, 1996.; F. Hélein, Applications harmoniques, lois des conservation et repère mobile, Diderotéditeur, Paris- New York-Amsterdam, 1996. |
| [9] | R. Hunt, On \(Lpq\); R. Hunt, On \(Lpq\) · Zbl 0181.40301 |
| [10] | E. Stein, G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, 1970.; E. Stein, G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, 1970. · Zbl 0232.42007 |
| [11] | Wente, H., An existence theorem for surfaces of constant mean curvature, J. Math. Anal. Appl., 26, 318-344 (1969) · Zbl 0181.11501 |
| [12] | Wente, H., Large solutions to the volume constrained Plateau problem, Arch. Rat. Mech. Anal., 75, 59-77 (1980) · Zbl 0473.49029 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
