DOI QR코드

DOI QR Code

ASYMPTOTIC BEHAVIOR OF A-HARMONIC FUNCTIONS AND p-EXTREMAL LENGTH

  • Published : 2010.03.31

Abstract

We describe the asymptotic behavior of functions of the Royden p-algebra in terms of p-extremal length. We also prove that each bounded $\cal{A}$-harmonic function with finite energy on a complete Riemannian manifold is uniquely determined by the behavior of the function along p-almost every curve.

Keywords

$\cal{A}$-harmonic function;p-harmonic boundary;comparison principle;maximum principle;p-extremal length;p-almost every curve

References

  1. J. Heinonen, T. Kilpelainen, and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993.
  2. J. Hesse, A p-extremal length and p-capacity equality, Ark. Mat. 13 (1975), 131–144. https://doi.org/10.1007/BF02386202
  3. E. Hewitt and K. Stormberg, Real and Abstract Analysis, Springer-Verlag, New York, Heidelberg, Berlin, 1965.
  4. I. Holopainen, Rough isometries and p-harmonic functions with finite Dirichlet integral, Rev. Mat. Iberoamericana 10 (1994), no. 1, 143–176.
  5. S. W. Kim and Y. H. Lee, Rough isometry and energy finite solutions for Schrodinger operator on Riemannian manifolds, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), no. 4, 855–873. https://doi.org/10.1017/S0308210500002717
  6. 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
  7. 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.
  8. S. Rickman, Quasiregular Mappings, Ergebnisse Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin, 1993.
  9. L. Sario and M. Nakai, Classification Theory of Riemann Surfaces, Springer Verlag, Berlin, Heidelberg, New York, 1970.
  10. H. Tanaka, Harmonic boundaries of Riemannian manifolds, Nonlinear Anal. 14 (1990), no. 1, 55–67. https://doi.org/10.1016/0362-546X(90)90135-4
  11. J. Vaisala, Lectures on n-Dimensional Quasiconformal Mappings, Lecture Notes in Math. 229 Springer-Verlag, Berlin, Heidelberg, New York, 1971.
  12. W. P. Ziemer, Extremal length and p-capacity, Michigan Math. J. 16 (1969), 43–51. https://doi.org/10.1307/mmj/1029000164
  13. W. P. Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation, Graduate Texts in Mathematics, 120. Springer-Verlag, New York, 1989.

Cited by

  1. Enantioselective α-hydrazination of α-fluoro-β-ketoesters catalyzed by chiral nickel complexes vol.130, pp.2, 2009, https://doi.org/10.1016/j.jfluchem.2008.11.001