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The sphere covering inequality and its applications

  • Changfeng Gui
  • Amir Moradifam
Article

Abstract

In this paper, we show that the total area of two distinct surfaces with Gaussian curvature equal to 1, which are also conformal to the Euclidean unit disk with the same conformal factor on the boundary, must be at least \(4 \pi \). In other words, the areas of these surfaces must cover the whole unit sphere after a proper rearrangement. We refer to this lower bound of total area as the Sphere Covering Inequality. The inequality and its generalizations are applied to a number of open problems related to Moser–Trudinger type inequalities, mean field equations and Onsager vortices, etc, and yield optimal results. In particular, we prove a conjecture proposed by Chang and Yang (Acta Math 159(3–4):215–259, 1987) in the study of Nirenberg problem in conformal geometry.

Notes

Acknowledgements

The authors would like to thank Professors Danielle Bartolucci, Alice Chang, Nassif Ghoussoub, Aleks Jevnikar, Yanyan Li, Fernando Marques, Richard Schoen, Paul Yang, etc for their interests and helpful comments on the earlier draft of the paper. The authors would also like to thank the anonymous referees for thoroughly checking the details of the manuscript and making valuble suggestions on the revision. The first author is partially supported by NSF Grant DMS-1601885. The second author is supported in part by NSF Grant DMS-1715850.

References

  1. 1.
    Aubin, T.: Problèmes isopérimétriques et espaces de Sobolev. J. Differ. Geom. 11(4), 573–598 (1976)CrossRefzbMATHGoogle Scholar
  2. 2.
    Aubin, T.: Meilleures constantes dans le théorème d’inclusion de Sobolev et un théorème de Fredholm non linéaire pour la transformation conforme de la courbure scalaire. J. Funct. Anal. 32(2), 148–174 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bandle, C.: Isoperimetric Inequalities and Applications, Volume 7 of Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston (1980)Google Scholar
  4. 4.
    Bartolucci, D., De Marchis, F.: Supercritical mean field equations on convex domains and the Onsager’s statistical description of two-dimensional turbulence. Arch. Ration. Mech. Anal. 217(2), 525–570 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bartolucci, D., Lin, C.-S.: Existence and uniqueness for mean field equations on multiply connected domains at the critical parameter. Math. Ann. 359(1–2), 1–44 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bartolucci, D., Lin, C.-S., Tarantello, G.: Uniqueness and symmetry results for solutions of a mean field equation on \(\mathbb{S}^2\) via a new bubbling phenomenon. Commun. Pure Appl. Math. 64(12), 1677–1730 (2011)CrossRefzbMATHGoogle Scholar
  7. 7.
    Beckner, W.: Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality. Ann. Math. (2) 138(1), 213–242 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bers, L.: Local behavior of solutions of general linear elliptic equations. Commun. Pure Appl. Math. 8, 473–496 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bol, G.: Isoperimetrische Ungleichungen für Bereiche auf Flächen. Jber. Deutsch. Math. Verein. 51, 219–257 (1941)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Brezis, H., Merle, F.: Uniform estimates and blow-up behavior for solutions of \(-\Delta u=V(x)e^u\) in two dimensions. Commun. Partial Differ. Equ. 16(8–9), 1223–1253 (1991)CrossRefzbMATHGoogle Scholar
  11. 11.
    Caglioti, E., Lions, P.-L., Marchioro, C., Pulvirenti, M.: A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. Commun. Math. Phys. 143(3), 501–525 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Caglioti, E., Lions, P.-L., Marchioro, C., Pulvirenti, M.: A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. II. Commun. Math. Phys. 174(2), 229–260 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chang, K.C., Liu, J.: On Nirenberg’s problem. Int. J. Math. 4(1), 35–58 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chang, S.-Y.A., Chen, C.-C., Lin, C.-S.: Extremal functions for a mean field equation in two dimension. In: Chang, S.-Y.A., Lin, C.-S., Yau, H.-T. (eds.) Lectures on Partial Differential Equations, Volume 2 of New Studies in Advanced Mathematics, pp. 61–93. International Press, Somerville (2003)Google Scholar
  15. 15.
    Chang, S.-Y.A., Yang, P.C.: Conformal deformation of metrics on \(S^2\). J. Differ. Geom. 27(2), 259–296 (1988)CrossRefGoogle Scholar
  16. 16.
    Chang, S.-Y.A., Yang, P.C.: Prescribing Gaussian curvature on \(S^2\). Acta Math. 159(3–4), 215–259 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Chanillo, S., Kiessling, M.: Rotational symmetry of solutions of some nonlinear problems in statistical mechanics and in geometry. Commun. Math. Phys. 160(2), 217–238 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Chen, R.M., Guo, Y., Spirn, D.: Asymptotic behavior and symmetry of condensate solutions in electroweak theory. J. Anal. Math. 117, 47–85 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Chen, W.X., Li, C.: Classification of solutions of some nonlinear elliptic equations. Duke Math. J. 63(3), 615–622 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Chen, W.X., Li, C.: A necessary and sufficient condition for the Nirenberg problem. Commun. Pure Appl. Math. 48(6), 657–667 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Cheng, K.-S., Lin, C.-S.: On the asymptotic behavior of solutions of the conformal Gaussian curvature equations in \({ R}^2\). Math. Ann. 308(1), 119–139 (1997)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Cheng, S.Y.: Eigenfunctions and eigenvalues of Laplacian. Amer. Math. Soc. Proc. Symp. Pure Math. 27(Part II), 185–193 (1975)Google Scholar
  23. 23.
    Dolbeault, J., Esteban, M.J., Jankowiak, G.: The Moser–Trudinger–Onofri inequality. Chin. Ann. Math. Ser. B 36(5), 777–802 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Dolbeault, J., Esteban, M.J., Tarantello, G.: Multiplicity results for the assigned Gauss curvature problem in \(\mathbb{R}^2\). Nonlinear Anal. 70(8), 2870–2881 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Feldman, J., Froese, R., Ghoussoub, N., Gui, C.: An improved Moser–Aubin–Onofri inequality for axially symmetric functions on \(S^2\). Calc. Var. Partial Differ. Equ. 6(2), 95–104 (1998)zbMATHGoogle Scholar
  26. 26.
    Ghoussoub, N., Lin, C.-S.: On the best constant in the Moser–Onofri–Aubin inequality. Commun. Math. Phys. 298(3), 869–878 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Ghoussoub, N., Moradifam, A.: Functional Inequalities: New Perspectives and New Applications, vol. 187. American Mathematical Society, Providence (2013)zbMATHGoogle Scholar
  28. 28.
    Gui, C., Moradifam, A.: Symmetry of solutions of a mean field equation on flat torus. Int. Math. Res. Not. (2016).  https://doi.org/10.1093/imrn/rnx121
  29. 29.
    Gui, C., Moradifam, A.: Uniqueness of solutions of mean field equations in \({ R}^2\). Proc. Am. Math. Soc. 146(3), 1231–1242 (2018)CrossRefzbMATHGoogle Scholar
  30. 30.
    Gui, C., Wei, J.: On a sharp Moser–Aubin–Onofri inequality for functions on \(S^2\) with symmetry. Pac. J. Math. 194(2), 349–358 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Hartman, P., Wintner, A.: On the local behavior of solutions of non-parabolic partial differential equations. Am. J. Math. 75, 449–476 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Kazdan, J., Warner, F.: Curvature functions for compact 2-manifolds. Ann. Math. 99, 14–47 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kiessling, M.K.-H.: Statistical mechanics of classical particles with logarithmic interactions. Commun. Pure Appl. Math. 46(1), 27–56 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Li, Y.Y.: Harnack type inequality: the method of moving planes. Commun. Math. Phys. 200(2), 421–444 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Lin, C.-S.: Topological degree for mean field equations on \(S^2\). Duke Math. J. 104(3), 501–536 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Lin, C.-S.: Uniqueness of solutions to the mean field equations for the spherical Onsager vortex. Arch. Ration. Mech. Anal. 153(2), 153–176 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Lin, C.-S., Lucia, M.: One-dimensional symmetry of periodic minimizers for a mean field equation. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6(2), 269–290 (2007)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1971)Google Scholar
  39. 39.
    Nehari, Z.: On the principal frequency of a membrane. Pac. J. Math. 2, 285–293 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Onofri, E.: On the positivity of the effective action in a theory of random surfaces. Commun. Math. Phys. 86(3), 321–326 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Osgood, B., Phillips, R., Sarnak, P.: Extremals of determinants of Laplacians. J. Funct. Anal. 80(1), 148–211 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Poliakovsky, A., Tarantello, G.: On a planar Liouville-type problem in the study of selfgravitating strings. J. Differ. Equ. 252(5), 3668–3693 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Suzuki, T.: Global analysis for a two-dimensional elliptic eigenvalue problem with the exponential nonlinearity. Ann. Inst. H. Poincaré Anal. Non Linéaire 9(4), 367–397 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Tarantello, G.: Analytical, geometrical and topological aspects of a class of mean field equations on surfaces. Discrete Contin. Dyn. Syst. 28(3), 931–973 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Trudinger, N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Yang, Y.S.: Self-duality of the gauge field equations and the cosmological constant. Commun. Math. Phys. 162(3), 481–498 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Mathematics and EconometricsHunan UniversityChangshaChina
  2. 2.Department of MathematicsUniversity of Texas at San AntonioSan AntonioUSA
  3. 3.Department of MathematicsUniversity of CaliforniaRiversideUSA

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