zbMATH — the first resource for mathematics

Convergence of surfaces of prescribed mean curvature. (English) Zbl 0696.49070
Let \(\Omega\) be the unit disk in \({\mathbb{R}}^ 2\), \(\Omega =\{(\alpha,\beta)\in {\mathbb{R}}^ 2|\) \(\alpha^ 2+\beta^ 2<1\}\), let \(\Gamma\) be a Jordan curve included in a ball of radius R of \({\mathbb{R}}^ 3\), and \(H\in C^{\infty}({\mathbb{R}}^ 3,{\mathbb{R}})\). We are searching for a function u: \({\bar \Omega}\to {\mathbb{R}}^ 3\) such that \[ (1)\quad \Delta u=2H(u)u_{\alpha}\wedge u_{\beta}\quad on\quad \Omega \] satisfying \[ | u_{\alpha}|^ 2-| u_{\beta}|^ 2=u_{\alpha}\cdot u_{\beta}=0\quad on\quad \Omega, \] \(u|_{\partial\Omega}:\;\partial\Omega \to \Gamma\) is a parametrization of \(\Gamma.\) This is Plateau’s problem. The author investigates the behavior of sequences of solutions of (1), when H is not constant. He deals with a sequence of solutions \((u^ n)\) converging weakly in \(H^ 1(\Omega)\). The main result is the fact that, for a subsequence, one has the convergence in \(H^ 1_{loc}(\Omega \setminus S)\) where S is a subset of finite cardinal, included in \(\Omega\). The author proves that the weak limit is a solution of (1). Such a result has already been obtained by J. Sacks and K. Uhlenbeck [Ann. Math., II. Ser. 113, 1-24 (1981; Zbl 0462.58014)] for harmonic maps of \(S^ 2\) into compact Riemannian manifolds.
The author also investigates the behavior of the sequence near a point of S and shows that, at least, one solution of \(\Delta u=2H(u)u_{\alpha}\wedge u_{\beta}\) on \({\mathbb{R}}^ 2\) can be “found” at every point of S. Finally the paper studies solutions of (1) on \({\mathbb{R}}^ 2\) and shows that their energy \(\int_{{\mathbb{R}}^ 2}| \nabla u|^ 2\) is minored by a positive constant.
Reviewer: T.Rassias

49Q05 Minimal surfaces and optimization
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
Full Text: DOI
[1] Brezis, H.; Coron, J.M., Multiple solutions of H-systems and Rellich’s conjecture, Communs pure appl. math., 37, 149-187, (1984) · Zbl 0537.49022
[2] Brezis, H.; Coron, J.M., Convergence of H-systems or how to blow bubble, Archs ration. mech. analysis, 89, 21-56, (1985) · Zbl 0584.49024
[3] Gilbarg D. & Trudinger N.S., Elliptic partial differential equations of second order, No. 224. Springer. · Zbl 1042.35002
[4] Gulliver, R., On the nonexistence of a hypersurface of prescribed Mean curvature with given boundary, Manuscripta math., 11, 15-39, (1974) · Zbl 0266.53002
[5] Heinz, E., On the nonexistence of a surface of constant Mean curvature with finite area and prescribed rectifiable boundary, Archs ration. mech. analysis, 35, (1969) · Zbl 0184.32802
[6] Heinz, E., Uber die regularität schwarcher lösungen nicht linear elliptisher systeme, Nachr. akad. wiss. gottingen II mathematisch physikalische klasse, 1, (1986)
[7] Hildebrandt, S., On the plateau problem for surfaces of constant Mean curvature, Communs pure appl. math., 23, 97-114, (1970) · Zbl 0181.38703
[8] Hildebrandt, S., Randwert probleme für flachen mit vorgeschrieben mittler krümmun und anverdungun auf die kapilaritäts theorie, Math. Z., 112, 205-213, (1969)
[9] Morrey, C., Multiple integrals in the calculus of variations, (1966), Springer Berlin · Zbl 0142.38701
[10] Sacks, J.; Uhlenbeck, K., The existence of minimal immersions of 2-spheres, Ann. math., 113, 1-24, (1981) · Zbl 0462.58014
[11] Schoen, R., Analytic aspects for the harmonic map problem, Math. sci. res. inst. publs, 2, (1984), Springer Berlin · Zbl 0551.58011
[12] Struwe, M., Non uniqueness in the plateau problem for surfaces of constant Mean curvature, Archs. ration. mech. analysis, 93, 135-157, (1986) · Zbl 0603.49027
[13] Struwe, M., Large H-surfaces via the mountain pass lemma, Math. annln, 270, 441-459, (1985) · Zbl 0582.58010
[14] Wente, H., Large solutions to the volume constrained plateau problem, Archs ration. mech. analysis, 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. 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.