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Free Boundaries on Two-Dimensional Cones

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Abstract

We study two phase problems posed over a two-dimensional cone generated by a smooth curve \(\gamma \) on the unit sphere. We show that when \(length(\gamma )<2\pi \) the free boundary avoids the vertex of the cone. When \(length(\gamma )\ge 2\pi \) we provide examples of minimizers such that the vertex belongs to the free boundary.

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Acknowledgments

M. A. was partially supported by NSF grants DMS-1101139 and DMS-1303632.

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Correspondence to Héctor Chang Lara.

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Communicated by Michael Taylor.

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Allen, M., Lara, H.C. Free Boundaries on Two-Dimensional Cones. J Geom Anal 25, 1547–1575 (2015). https://doi.org/10.1007/s12220-014-9484-3

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