Advertisement

Minimal graphs in the hyperbolic space with singular asymptotic boundaries

  • Qing Han
  • Weiming Shen
  • Yue WangEmail author
Article
  • 75 Downloads

Abstract

We study asymptotic behaviors of solutions f to the Dirichlet problem for minimal graphs in the hyperbolic space with singular asymptotic boundaries under the assumption that the boundaries are piecewise regular with positive curvatures, a case which also arises in the study of a Chaplygin gas. We derive an estimate of such solutions by the corresponding solutions in the intersections of interior tangent balls. The positivity of curvatures plays an important role.

Mathematics Subject Classification

35J93 

Notes

References

  1. 1.
    Han, Q., Jiang, X.: Boundary Expansions for Minimal Graphs in the Hyperbolic Space. arxiv:1412.7608
  2. 2.
    Han, Q., Shen, W.: Boundary expansions for Liouville’s equation in planar singular domains. J. Funct. Anal. 274, 1790–1824 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Han, Q., Shen, W.: The Loewner–Nirenberg problem in singular domains. arxiv:1511.01146v1
  4. 4.
    Han, Q., Shen, W., Wang, Y.: Optimal regularity of minimal graphs in the hyperbolic space. Calc. Var. Partial Differ. Equ. 55, no. 1, Art. 3, 19pp (2016)Google Scholar
  5. 5.
    Lin, F.-H.: On the Dirichlet problem for minimal graphs in hyperbolic space. Invent. Math. 96, 593–612 (1989)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Serre, D.: Multidimensional shock interaction for a Chaplygin gas. Arch. Ration. Mech. Anal. 191, 539–577 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Shen, W., Wang, Y.: Refined asymptotics for minimal graphs in the hyperbolic space. Sci. China Math. 62, 381–390 (2019)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Notre DameNotre DameUSA
  2. 2.School of Mathematical SciencesCapital Normal UniversityBeijingChina

Personalised recommendations