×

Symmetric boundary values for the Dirichlet problem for harmonic maps from the disc into the 2-sphere. (English) Zbl 1131.58013

The energy of mappings \(D^2\to S^2\) is invariant under the action of \(\text{Aut}(D)\times O(3)\) acting on \(D^2\times S^2\), where \(\text{Aut}(D)\) is the group of biholomorphic diffeomorphisms of \(D^2\) to itself. Given Dirichlet boundary data \(\gamma:\partial D\to S^2\) are invariant under some (maybe trivial) subgroup of \(\text{Aut}(D)\times O(3)\). The possible stabilizer subgroups of boundary data in the natural classes \(H^{1/2}(S^1,S^2)\) and \(C^0(S^1,S^2)\) are classified in the paper.
In the positively oriented part \(\text{Aut}(G)^+\times SO(3)\), the possible stabilizers of nonconstant data are conjugate to a rotation group generated by a rotation of some angle \(\alpha\) in the first factor and of some angle \(n\alpha\) (\(n\in \mathbb N\)) in the second factor. Note that this group may be finite or infinite depending on whether \(\alpha/\pi\) is rational.
As an application of the symmetry considerations, the author proves existence of nonsymmetric Dirichlet data for which there are at least two distinct solutions belonging to the same homotopy class. Also, he shows that there are data which allow a continuum of distinct solutions.

MSC:

58E20 Harmonic maps, etc.
35G30 Boundary value problems for nonlinear higher-order PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Beardon A.F. (1983). The geometry of discrete groups. Graduate Texts in Mathematics, vol. 91. Springer, Berlin · Zbl 0528.30001
[2] Berenstein, C.A., Gay, R.: Complex variables. Graduate Texts in Mathematics, vol. 125. Springer, Berlin (1991)(An introduction) · Zbl 0741.30001
[3] Bethuel F. and Demengel F. (1995). Extensions for Sobolev mappings between manifolds. Calc. Var. Partial Differ. Equ. 3(4): 475–491 · Zbl 0846.46021 · doi:10.1007/BF01187897
[4] Brezis, H.: New questions related to the topological degree. In: The unity of mathematics. Progr. Math., vol. 244, pp. 137–154. Birkhäuser Boston, Boston (2006) · Zbl 1109.58017
[5] Brezis H. and Coron J.-M. (1983). Large solutions for harmonic maps in two dimensions. Commun. Math. Phys. 92(2): 203–215 · Zbl 0532.58006 · doi:10.1007/BF01210846
[6] Brezis H. and Nirenberg L. (1995). Degree theory and BMO. I. Compact manifolds without boundaries. Selecta Math. (N.S.) 1(2): 197–263 · Zbl 0852.58010 · doi:10.1007/BF01671566
[7] Hall, M. Jr.: The Theory of Groups. Chelsea Publishing Co., New York (1976) (Reprinting of the 1968 edition)
[8] Hélein F. (1991). Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne. C. R. Acad. Sci. Paris Sér. I Math. 312(8): 591–596
[9] Jost J. (1984). The Dirichlet problem for harmonic maps from a surface with boundary onto a 2-sphere with nonconstant boundary values. J. Differ. Geom. 19(2): 393–401 · Zbl 0551.58012
[10] Kuwert E. (1994). Minimizing the energy of maps from a surface into a 2-sphere with prescribed degree and boundary values. Manuscr. Math. 83(1): 31–38 · Zbl 0808.58017 · doi:10.1007/BF02567598
[11] Lemaire L. (1978). Applications harmoniques de surfaces riemanniennes. J. Differ. Geom. 13(1): 51–78 · Zbl 0388.58003
[12] Lyndon R.C. and Ullman J.L. (1967). Groups of elliptic linear fractional transformations. Proc. Am. Math. Soc. 18: 1119–1124 · Zbl 0163.32202 · doi:10.1090/S0002-9939-1967-0222182-8
[13] Pierre M. (2005). Symmetry breaking for the Dirichlet problem for harmonic maps from the disc into the 2-sphere. Adv. Differ. Equ. 10(6): 675–694 · Zbl 1101.58014
[14] Qing J. (1992). Multiple solutions of the Dirichlet problem for harmonic maps from discs into 2-spheres. Manuscr. Math. 77(4): 435–446 · Zbl 0788.58018 · doi:10.1007/BF02567066
[15] Qing J. (1992). Remark on the Dirichlet problem for harmonic maps from the disc into the 2-sphere. Proc. R. Soc. Edinb. Sect. A 122(1–2): 63–67 · Zbl 0769.58014
[16] Schoen R. and Uhlenbeck K. (1983). Boundary regularity and the Dirichlet problem for harmonic maps. J. Differ. Geom. 18(2): 253–268 · Zbl 0547.58020
[17] Soyeur A. (1989). The Dirichlet problem for harmonic maps from the disc into the 2-sphere. Proc. R. Soc. Edinb. Sect. A 113(3-4): 229–234 · Zbl 0701.58020
[18] Weyl H. (1952). Symmetry. Princeton University Press, Princeton · Zbl 0046.00406
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.