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    188宝金博页面版: Existence of stable H-surfaces in cones and their representation as radial graphs

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    内容提示： Calc. Var. (2016) 55:131DOI 10.1007/s00526-016-1074-8 Calculus of VariationsExistence of stable H-surfaces in cones and theirrepresentation as radial graphsPaolo Caldiroli 1 · Alessandro Iacopetti 1Received: 11 December 2015 / Accepted: 5 October 2016 / Published online: 27 October 2016? Springer-Verlag Berlin Heidelberg 2016Abstract InthispaperwestudythePlateauproblemfordisk-typesurfacescontainedinconicregions of R 3 and with prescribed mean curvature H. Assuming a suitable growth conditionon H,weprovee...

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    Calc. Var. (2016) 55:131DOI 10.1007/s00526-016-1074-8 Calculus of VariationsExistence of stable H-surfaces in cones and theirrepresentation as radial graphsPaolo Caldiroli 1 · Alessandro Iacopetti 1Received: 11 December 2015 / Accepted: 5 October 2016 / Published online: 27 October 2016© Springer-Verlag Berlin Heidelberg 2016Abstract InthispaperwestudythePlateauproblemfordisk-typesurfacescontainedinconicregions of R 3 and with prescribed mean curvature H. Assuming a suitable growth conditionon H,weproveexistenceofaleastenergy H-surface X spanninganarbitraryJordancurve?taken in the cone. Then we address the problem of describing such surface X as radial graphwhen the Jordan curve ? admits a radial representation. Assuming a suitable monotonicitycondition on the mapping λ ?→ λH(λp) and some strong convexity-type condition on theradial projection of the Jordan curve ?, we show that the H-surface X can be represented asa radial graph.Mathematics Subject Classif i cation 53A10 · 35J66 · 57R401 IntroductionIn the present paper we aim to investigate some aspects on the Plateau problem for disk-typesurfaces with prescribed mean curvature in the directions described as follows. Fixing a coneof angular radius βC β := {p = (x, y,z) ∈ R 3 | z > |p|cosβ},a Jordan curve ? ⊂ C β \{0}, and a mapping H : C β → R , we are interested in f i ndingconditions on H, possibly related to β, ensuring that stable surfaces in C β \{0} with meancurvature H, spanning λ? do exist for every λ > 0. Moreover we address the problem ofdescribingsuchsurfacesasradialgraphswhentheirboundariesadmitaradialrepresentation.Communicated by A. Malchiodi.B Alessandro Iacopettialessandro.iacopetti@unito.itPaolo Caldirolipaolo.caldiroli@unito.it1Dipartimento di Matematica, Università di Torino, via Carlo Alberto, 10, 10123 Torino, Italy123

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