# zbMATH — the first resource for mathematics

The Dirichlet problem for harmonic maps from the disk into the Euclidean n-sphere. (English) Zbl 0597.35022
Let $$\Omega =\{(x,y)\in {\mathbb{R}}^ 2:$$ $$x^ 2+y^ 2<1\}$$, $$S^ n=\{v\in {\mathbb{R}}^{n+1}:$$ $$| v| =1\}$$, $$n\geq 2$$, $$\gamma \in C^{2,\delta}(\partial \Omega,S^ n)$$ a nonconstant function $$(0<\delta <1)$$, $$\Sigma_ p=\{\sigma \in C^ 0(S^{n- 2},W_{\gamma}^{1,p}(\Omega,S^ n)),$$ $$\sigma$$ is not homotopic to a constant$$\}$$ where $$p>2$$, $$W_{\gamma}^{1,p}(\Omega,S^ n)=\{u\in W^{1,p}(\Omega,S^ n):\quad u=\gamma \quad on\quad \partial \Omega \},$$ $$\Sigma =\cup_{p>2}\Sigma_ p$$ and $$c=\inf_{\sigma \in \Sigma}(\max_{s\in S^{n-2}}E(\sigma (s))$$, where $$E(u)=\int_{\Omega}| \nabla u|^ 2 dx$$. Then:
Theorem. There exists at least $$u\in C^{2,\delta}({\bar \Omega},S^ n)$$ such that $$E(u)=c$$, $$-\Delta u=u| \nabla u|^ 2$$, $$u|_{\partial \Omega}=\gamma$$. Moreover if $$c=m$$ there exist infinitely many u when $$n\geq 3$$ (and at least two when $$n=2)$$, where $$m=\inf \{E(u):\quad u\in H^ 1(\Omega,{\mathbb{R}}^{n+1}),\quad u|_{\partial \Omega}=\gamma,\quad | u| =1\quad a.e.\}.$$
Reviewer: G.Bottaro

##### MSC:
 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 31A05 Harmonic, subharmonic, superharmonic functions in two dimensions 58E12 Variational problems concerning minimal surfaces (problems in two independent variables) 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
Full Text:
##### References:
  Aubin, Th, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures et Appl., t. 55, 269-296, (1976) · Zbl 0336.53033  Brezis, H.; Coron, J. M., Multiple solutions of H-systems and rellich’s conjecture, Comm. Pure Appl. Math., t. XXXVII, 149-187, (1984) · Zbl 0537.49022  Brezis, H.; Coron, J. M., Large solutions for harmonic maps in two dimensions, Comm. Math. Phys., t. 92, 203-215, (1983) · Zbl 0532.58006  Brezis, H.; Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math, t. XXXVI, 437-477, (1983) · Zbl 0541.35029  Calabi, E., Minimal immersions of surfaces in Euclidean spheres, J. Diff. Geometry, t. 1, 111-125, (1967) · Zbl 0171.20504  Gulliver, R. D.; Osserman, R.; Royden, H. L., A theory of branched immersions of surfaces, Amer. J. Math., t. 95, 750-812, (1973) · Zbl 0295.53002  Gilbarg, D.; Trudinger, N. S., Elliptic partial differential equations of second order, (1977), Springer-Verlag Berlin-Heidelberg-New York · Zbl 0691.35001  Hildebrandt, S.; Widman, K. O., Some regularity results for quasilinear elliptic systems of second order, Math. Z., t. 142, 67-86, (1975) · Zbl 0317.35040  J. Jost, The Dirichlet problem for harmonic maps from a surface with boundary onto a 2-sphere with nonconstant boundary values. J. Diff. Geometry (to appear). · Zbl 0551.58012  Ladyzenskaya, O. A.; Ural’ceva, N. N., Linear and quasilinear elliptic equations, (1968), Academic Press New York and London  Ladyzenskaya, O. A.; Ural’ceva, N. N., Linear and quasilinear elliptic equations, (1973), Nauka Moscow  Lemaire, L., Applications harmoniques de surfaces riemanniennes, J. Diff. Geometry, t. 13, 51-78, (1978) · Zbl 0388.58003  E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Annals of Math. (to appear). · Zbl 0527.42011  Lions, P. L., The concentration compactness principle in the calculus of variations the limit case, Riv. Iberoamericana (to appear) and Comptes Rendus Acad. Sc. Paris, t. 296, série I, 645-648, (1983)  Morrey, C. B., On the solutions of quasilinear elliptic partial differential equations, Trans. Amer. Math. Soc., t. 43, 126-166, (1938) · JFM 64.0460.02  Morrey, C. B., Multiple integrals in the calculus of variations, (1966), Springer-Verlag Berlin-Heidelberg-New York · Zbl 0142.38701  Nirenberg, L., On nonlinear elliptic partial differential equations and Hölder continuity, Comm. Pure App. Math., t. 6, 103-156, (1953) · Zbl 0050.09801  Sacks, J.; Uhlenbeck, K., The existence of minimal immersions of 2-spheres, Annals of Math., t. 113, 1-24, (1981) · Zbl 0462.58014  Schoen, R.; Uhlenbeck, K., Boundary regularity and the Dirichlet problem for harmonic maps, J. Diff. Geometry, t. 18, 253-268, (1983) · Zbl 0547.58020  M. Struwe, Nonuniqueness in the Plateau problem for surfaces of constant mean curvature (to appear). · Zbl 0603.49027  C. Taubes, The existence of a non-minimal solution to the SU(2) Yang-Mills-Higgs equations on ℝ^3 (to appear). · Zbl 0514.58016  Wente, H., The differential equation δx = 2Hx_u ∧ x_v with vanishing boundary values, Proc. A. M. S., t. 50, 131-137, (1975) · Zbl 0313.35030  Wente, H., The Dirichlet problem with a volume constraint, Manuscripta Math., t. 11, 141-157, (1974) · Zbl 0268.35031  Wiegner, M., A-priori schranken für Lösungen gewisser elliptischer systeme, Math. Z., t. 147, 21-28, (1976) · Zbl 0316.35039
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.