Gelfand problem on a large spherical cap

Abstract

We study the behaviour of the minimal solution to the Gelfand problem on a spherical cap under the Dirichlet boundary conditions. The asymptotic behaviour of the solution is discussed as the cap approaches the whole sphere. The results are based on the sharp estimate of the torsion function of the spherical cap in terms of the principle eigenvalue which we derive in this work.

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Acknowledgements

The authors are grateful to the anonymous referee for their comments which helped to improve the exposition in the paper. Part of this research was carried out while VM was visiting Osaka Prefecture University. VM thanks the Department of Mathematical Sciences for its support and hospitality. YK is supported in part by JSPS KAKENHI Grant Number 19K03588.

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Correspondence to Vitaly Moroz.

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Kabeya, Y., Moroz, V. Gelfand problem on a large spherical cap. J Elliptic Parabol Equ (2020). https://doi.org/10.1007/s41808-020-00091-9

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Keywords

  • Gelphand problem
  • Spherical cap
  • Torsion function
  • Eigenvalues
  • Spherical harmonics

Mathematics Subject Classification

  • 35J60
  • 33C55
  • 35R01