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On the isoperimetric problem for radial log-convex densities

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Abstract

Given a smooth, radial, uniformly log-convex density e V on \({\mathbb{R}^n}\) , n ≥ 2, we characterize isoperimetric sets E with respect to weighted perimeter \({\int_{\partial E}e^{V} d \mathcal{H}^{n-1}}\) and weighted volume m = ∫ E e V as balls centered at the origin, provided \({m \in [0, m_0)}\) for some (potentially computable) m 0>0; this affirmatively answers conjecture (Rosales et al. Calc Var Part Differ Equat 31(1):27–46, 2008, Conjecture 3.12) for such values of the weighted volume parameter. We also prove that the set of weighted volumes such that this characterization holds true is open, thus reducing the proof of the full conjecture to excluding the possibility of bifurcation values of the weighted volume parameter. Finally, we show the validity of the conjecture when V belongs to a C 2-neighborhood of c|x|2 (c> 0).

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Authors and Affiliations

  1. Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, TX, 78712, USA

    A. Figalli

  2. Dipartimento di Matematica “U. Dini”, Università di Firenze, Viale Morgagni 67/A, 50134, Firenze, Italy

    F. Maggi

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  1. A. Figalli
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  2. F. Maggi
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Correspondence to A. Figalli.

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Communicated by L.Ambrosio.

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Figalli, A., Maggi, F. On the isoperimetric problem for radial log-convex densities. Calc. Var. 48, 447–489 (2013). https://doi.org/10.1007/s00526-012-0557-5

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  • DOI: https://doi.org/10.1007/s00526-012-0557-5

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