Abstract
In this paper we study the Plateau problem for disk-type surfaces contained in conic regions of \(\mathbb R^{3}\) and with prescribed mean curvature H. Assuming a suitable growth condition on H, we prove existence of a least energy H-surface X spanning an arbitrary Jordan curve \(\Gamma \) taken in the cone. Then we address the problem of describing such surface X as radial graph when the Jordan curve \(\Gamma \) admits a radial representation. Assuming a suitable monotonicity condition on the mapping \(\lambda \mapsto \lambda H(\lambda p)\) and some strong convexity-type condition on the radial projection of the Jordan curve \(\Gamma \), we show that the H-surface X can be represented as a radial graph.
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Acknowledgments
Research partially supported by the project ERC Advanced Grant 2013 n. 339958 Complex Patterns for Strongly Interacting Dynamical Systems COMPAT, and by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The first author is also supported by the PRIN-2012-74FYK7 Grant “Variational and perturbative aspects of nonlinear differential problems”
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Communicated by A. Malchiodi.
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Caldiroli, P., Iacopetti, A. Existence of stable H-surfaces in cones and their representation as radial graphs. Calc. Var. 55, 131 (2016). https://doi.org/10.1007/s00526-016-1074-8
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DOI: https://doi.org/10.1007/s00526-016-1074-8