Multiple solutions to the planar plateau problem. (English) Zbl 1265.53067

The author studies the existence and multiplicity of planar simple curves with fixed ends and prescribed geodesic curvature. This problem can be phrased in terms of the Dirichlet problem: \[ \ddot \gamma =| \dot {\gamma }| k(\gamma ,t)J\dot \gamma \quad \text{in } [0,1],\quad \gamma (0)=(a,0),\quad \gamma (1)=(-a,0), \] where the unknown function \(\gamma \) is in \(C^{2}([0,1],\mathbb {R}^{2})\), \(J\) denotes the rotation by \(\pi /2\), and the data are a positive number \(a\) and the mapping \(k\in C(\mathbb {R}^{2}\times [0,1],\mathbb {R})\). Let \(\underline {k}=\inf _{\mathbb {R}^{2}\times [0,1]}k\) and \(\overline {k}=\sup _{\mathbb {R}^{2}\times [0,1]}k\). The author proves that if \(0<\underline {k}\leq \overline {k}<a^{-1}\) then there exists a simple curve parametrized by a solution of the above problem. If, in addition, \(\overline {k}<\underline {k}(1+a\overline {k})\) then there exist two different simple curves parametrized by solutions of the above problem. Notice that the assumptions are satisfied if \(a^{-1}/2<\underline {k}\leq \overline {k}<a^{-1}\). The result is obtained as an application of the Leray-Schauder degree theory, matched with suitable a priori estimates. The statement concerning the existence of two solutions is rather interesting and it seems to be the first non perturbative multiplicity result for geometrical problems of this kind, with prescribed curvature.


53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A04 Curves in Euclidean and related spaces
34L30 Nonlinear ordinary differential operators
53C22 Geodesics in global differential geometry
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