Download PDF - The boundary Harnack principle for the fractional LaplacianAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

The boundary Harnack principle for the fractional Laplacian

Authors 1

Affiliations

  1. Institute of Mathematics, Technical University of Wrocław, 50-370 Wrocław, Poland

Abstract

We study nonnegative functions which are harmonic on a Lipschitz domain with respect to symmetric stable processes. We prove that if two such functions vanish continuously outside the domain near a part of its boundary, then their ratio is bounded near this part of the boundary.

Keywords

ENG
boundary Harnack principle, symmetric stable processes, harmonic functions, Lipschitz domains

Bibliography

  1. R. F. Bass, Probabilistic Techniques in Analysis, Springer, New York, 1995.
  2. R. F. Bass and K. Burdzy, A probabilistic proof of the boundary Harnack principle, in: E. Çinlar, K. L. Chung, and R. K. Getoor (eds.), Seminar on Stochastic Processes, 1989, Birkhäuser, Boston, 1990, 1-16.
  3. R. F. Bass and K. Burdzy, The boundary Harnack principle for non-divergence form elliptic operators, J. London Math. Soc. 50 (1994), 157-169.
  4. R. F. Bass and M. Cranston, Exit times for symmetric stable processes in Rn, Ann. Probab. 11 (1983), 578-588.
  5. R. M. Blumenthal and R. K. Getoor, Markov Processes and Their Potential Theory, Pure and Appl. Math., Academic Press, New York, 1968.
  6. K. Burdzy, Multidimensional Brownian Excursions and Potential Theory, Pitman Res. Notes in Math. 164, Longman, Harlow, 1987.
  7. L. Caffarelli, E. Fabes, S. Mortola, and S. Salsa, Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana Univ. Math. J. 30 (1981), 621-640.
  8. M. Cranston, E. Fabes, and Z. Zhao, Conditional gauge and potential theory for the Schrödinger operator, Trans. Amer. Math. Soc. 307 (1988), 171-194.
  9. B. Dahlberg, Estimates of harmonic measure, Arch. Rational Mech. Anal. 65 (1977), 275-288.
  10. E. B. Dynkin, Markov Processes, Vols. I, II, Academic Press, New York, 1965.
  11. D. S. Jerison and C. E. Kenig, Boundary behavior of harmonic functions in non-tangentially accessible domains, Adv. in Math. 46 (1982), 80-147.
  12. D. S. Jerison and C. E. Kenig, Boundary value problems on Lipschitz domains, in: W. Littman (ed.), Studies in Partial Differential Equations, MAA Stud. Math. 23, Math. Assoc. Amer., 1982, 1-68.
  13. N. S. Landkof, Foundations of Modern Potential Theory, Springer, New York, 1972.
  14. S. C. Port and C. J. Stone, Infinitely divisible processes and their potential theory, Ann. Inst. Fourier (Grenoble) 21 (2) (1971), 157-275; 21 (4) (1971), 179-265.
  15. S. Watanabe, On stable processes with boundary conditions, J. Math. Soc. Japan 14 (1962), 170-198.
  16. K. Yosida, Functional Analysis, Springer, New York, 1971.
Studia Mathematica
1997/123/1
Export to PBN, Opens in new tab
Pages:
43-80
Main language of publication
English
Received
1996-02-19
Published
1997
Exact and natural sciences