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Hypersurfaces with constant mean curvature and prescribed area

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Summary

We consider—in the setting of geometric measure theory—hypersurfacesT (of codimension one) with prescribed boundaryB in Euclideann+1 space which maximize volume (i.e.T together with a fixed hypersurfaceT 0 encloses oriented volume) subject to a mass constraint. We prove existence and optimal regularity of solutionsT of such variational problems and we show that, on the regular part of its support,T is a classical hypersurface of constant mean curvature. We also prove that the solutionsT become more and more spherical as the valuem of the mass constraint approaches ∞.

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Authors and Affiliations

  1. Institut für Reine Mathematik, Humboldt-Universität zu Berlin, Sitz: Ziegelstr. 13A, 10099, Berlin

    Frank Duzaar

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  1. Frank Duzaar
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This work was done at the Centre for Mathematics and its Applications at the Australian National University, Canberra while the author was a visiting member

This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.

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Duzaar, F. Hypersurfaces with constant mean curvature and prescribed area. Manuscripta Math 91, 303–315 (1996). https://doi.org/10.1007/BF02567956

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  • DOI: https://doi.org/10.1007/BF02567956

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