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1997 | 123 | 1 | 43-80
Tytuł artykułu

The boundary Harnack principle for the fractional Laplacian

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EN
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EN
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.
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Bibliografia
  • [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 $R^n$, 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.
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Bibliografia
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bwmeta1.element.bwnjournal-article-smv123i1p43bwm
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