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UNIQUENESS OF SOLUTIONS FOR THE BOUNDARY VALUE PROBLEM OF CERTAIN NONLINEAR ELLIPTIC OPERATORS VIA p-HARMONIC BOUNDARY

  • Lee, Yong Hah
  • Received : 2016.11.07
  • Accepted : 2017.02.02
  • Published : 2017.10.31

Abstract

We prove the uniqueness of solutions for the boundary value problem of certain nonlinear elliptic operators in the setting: Given any continuous function f on the p-harmonic boundary of a complete Riemannian manifold, there exists a unique solution of certain nonlinear elliptic operators, which is a limit of a sequence of solutions of the operators with finite energy in the sense of supremum norm, on the manifold taking the same boundary value at each p-harmonic boundary as that of f.

Keywords

${\mathcal{A}}$-harmonic function;p-harmonic boundary;boundary value problem

References

  1. J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geometry 6 (1971), 119-128. https://doi.org/10.4310/jdg/1214430220
  2. E. Hewitt and K. Stormberg, Real and Abstract Analysis, Springer-Verlag, New York, Heidelberg, Berlin, 1965.
  3. Y. H. Lee, Rough isometry and energy finite solutions of elliptic equations on Riemannian manifolds, Math. Ann. 318 (2000), no. 1, 181-204. https://doi.org/10.1007/s002080000118
  4. Y. H. Lee, Rough isometry and p-harmonic boundaries of complete Riemannian manifolds, Potential Anal. 23 (2005), no.1, 83-97. https://doi.org/10.1007/s11118-004-3261-z
  5. J. Maly and W. P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, Mathematical Surveys and Monographs, 51. American Mathematical Society, Providence, RI, 1997.
  6. L. Sario and M. Nakai, Classification Theory of Riemann Surfaces, Springer Verlag, Berlin, Heidelberg, New York, 1970.

Acknowledgement

Supported by : National Research Foundation of Korea(NRF)