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A singular Sphere Covering Inequality: uniqueness and symmetry of solutions to singular Liouville-type equations

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Abstract

We derive a singular version of the Sphere Covering Inequality which was recently introduced in Gui and Moradifam (Invent Math. https://doi.org/10.1007/s00222-018-0820-2, 2018) suitable for treating singular Liouville-type problems with superharmonic weights. As an application we deduce new uniqueness results for solutions of the singular mean field equation both on spheres and on bounded domains, as well as new self-contained proofs of previously known results, such as the uniqueness of spherical convex polytopes first established in Luo and Tian (Proc Am Math Soc 116(4):1119–1129, 1992). Furthermore, we derive new symmetry results for the spherical Onsager vortex equation.

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Authors and Affiliations

  1. Department of Mathematics, University of Rome “Tor Vergata”, Via della Ricerca Scientifica 1, 00133, Rome, Italy

    Daniele Bartolucci

  2. Department of Mathematics, University of Texas at San Antonio, San Antonio, TX, USA

    Changfeng Gui

  3. School of Mathematics and Statistics, The Central South University, Changsha, China

    Changfeng Gui

  4. Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italy

    Aleks Jevnikar

  5. Department of Mathematics, University of California, Riverside, CA, USA

    Amir Moradifam

Authors
  1. Daniele Bartolucci
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  2. Changfeng Gui
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  3. Aleks Jevnikar
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  4. Amir Moradifam
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Corresponding author

Correspondence to Aleks Jevnikar.

Additional information

Communicated by Y. Giga.

D.B. and A.J. are partially supported by FIRB project “Analysis and Beyond”, by PRIN project 2012, ERC PE1_11, “Variational and perturbative aspects in nonlinear differential problems”, and by the Consolidate the Foundations project 2015 (sponsored by Univ. of Rome “Tor Vergata”), “Nonlinear Differential Problems and their Applications”. C.G. is partially supported by NSF Grant DMS-1601885, and A.M. is supported by NSF Grant DMS-1715850.

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Bartolucci, D., Gui, C., Jevnikar, A. et al. A singular Sphere Covering Inequality: uniqueness and symmetry of solutions to singular Liouville-type equations. Math. Ann. 374, 1883–1922 (2019). https://doi.org/10.1007/s00208-018-1761-1

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  • DOI: https://doi.org/10.1007/s00208-018-1761-1

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