The boundary Harnack principle for the fractional Laplacian

Krzysztof Bogdan

Studia Mathematica (1997)

  • Volume: 123, Issue: 1, page 43-80
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Bogdan, Krzysztof. "The boundary Harnack principle for the fractional Laplacian." Studia Mathematica 123.1 (1997): 43-80. <http://eudml.org/doc/216379>.

@article{Bogdan1997,
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.},
author = {Bogdan, Krzysztof},
journal = {Studia Mathematica},
keywords = {boundary Harnack principle; symmetric stable processes; harmonic functions; Lipschitz domains; Laplacian; fractional powers; symmetric stable semigroup; Riesz potentials},
language = {eng},
number = {1},
pages = {43-80},
title = {The boundary Harnack principle for the fractional Laplacian},
url = {http://eudml.org/doc/216379},
volume = {123},
year = {1997},
}

TY - JOUR
AU - Bogdan, Krzysztof
TI - The boundary Harnack principle for the fractional Laplacian
JO - Studia Mathematica
PY - 1997
VL - 123
IS - 1
SP - 43
EP - 80
AB - 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.
LA - eng
KW - boundary Harnack principle; symmetric stable processes; harmonic functions; Lipschitz domains; Laplacian; fractional powers; symmetric stable semigroup; Riesz potentials
UR - http://eudml.org/doc/216379
ER -

References

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  2. [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. [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. Zbl0806.35025
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  9. [9] B. Dahlberg, Estimates of harmonic measure, Arch. Rational Mech. Anal. 65 (1977), 275-288. Zbl0406.28009
  10. [10] E. B. Dynkin, Markov Processes, Vols. I, II, Academic Press, New York, 1965. 
  11. [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. Zbl0514.31003
  12. [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. Zbl0529.31007
  13. [13] N. S. Landkof, Foundations of Modern Potential Theory, Springer, New York, 1972. 
  14. [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. Zbl0195.47601
  15. [15] S. Watanabe, On stable processes with boundary conditions, J. Math. Soc. Japan 14 (1962), 170-198. Zbl0114.33504
  16. [16] K. Yosida, Functional Analysis, Springer, New York, 1971. 

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