Comparisons of kernel functions boundary Harnack principle and relative Fatou theorem on Lipschitz domains

Jang-Mei G. Wu

Annales de l'institut Fourier (1978)

  • Volume: 28, Issue: 4, page 147-167
  • ISSN: 0373-0956

Abstract

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On a Lipschitz domain D in R n , three theorems on harmonic functions are proved. The first (boundary Harnack principle) compares two positive harmonic functions at interior points near an open subset of the boundary where both functions vanish. The second extends some familiar geometric facts about the Poisson kernel on a sphere to the Poisson kernel on D . The third theorem, on non-tangential limits of quotient of two positive harmonic functions in D , generalizes Doob’s relative Fatou theorem on a sphere. The main tools are maximum principle, Harnack inequality and differentiation of measures.

How to cite

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Wu, Jang-Mei G.. "Comparisons of kernel functions boundary Harnack principle and relative Fatou theorem on Lipschitz domains." Annales de l'institut Fourier 28.4 (1978): 147-167. <http://eudml.org/doc/74378>.

@article{Wu1978,
abstract = {On a Lipschitz domain $D$ in $\{\bf R\}^n$, three theorems on harmonic functions are proved. The first (boundary Harnack principle) compares two positive harmonic functions at interior points near an open subset of the boundary where both functions vanish. The second extends some familiar geometric facts about the Poisson kernel on a sphere to the Poisson kernel on $D$. The third theorem, on non-tangential limits of quotient of two positive harmonic functions in $D$, generalizes Doob’s relative Fatou theorem on a sphere. The main tools are maximum principle, Harnack inequality and differentiation of measures.},
author = {Wu, Jang-Mei G.},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {4},
pages = {147-167},
publisher = {Association des Annales de l'Institut Fourier},
title = {Comparisons of kernel functions boundary Harnack principle and relative Fatou theorem on Lipschitz domains},
url = {http://eudml.org/doc/74378},
volume = {28},
year = {1978},
}

TY - JOUR
AU - Wu, Jang-Mei G.
TI - Comparisons of kernel functions boundary Harnack principle and relative Fatou theorem on Lipschitz domains
JO - Annales de l'institut Fourier
PY - 1978
PB - Association des Annales de l'Institut Fourier
VL - 28
IS - 4
SP - 147
EP - 167
AB - On a Lipschitz domain $D$ in ${\bf R}^n$, three theorems on harmonic functions are proved. The first (boundary Harnack principle) compares two positive harmonic functions at interior points near an open subset of the boundary where both functions vanish. The second extends some familiar geometric facts about the Poisson kernel on a sphere to the Poisson kernel on $D$. The third theorem, on non-tangential limits of quotient of two positive harmonic functions in $D$, generalizes Doob’s relative Fatou theorem on a sphere. The main tools are maximum principle, Harnack inequality and differentiation of measures.
LA - eng
UR - http://eudml.org/doc/74378
ER -

References

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  1. [1] A. ANCONA, Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine Lipschitzien, Ann. Inst. Fourier, (to appear). Zbl0377.31001
  2. [2] A. S. BESICOVITCH, A general form of the covering principle and relative differentiation of additive functions II, Proc. Cambridge Philos. Soc., 42 (1946), 1-10. Zbl0063.00353MR7,281e
  3. [3] M. BRELOT, Remarques sur les zéros à la frontière des fonctions harmoniques positives, Un. Mat. Ita., Boll., Suppl., Ser. 4, 12 (1975), 314-319. Zbl0338.31004MR54 #7823
  4. [4] M. BRELOT et J. L. DOOB, Limites angulaires et limites fines, Ann. Institut Fourier, 13, 2 (1963), 395-415. Zbl0132.33902MR33 #4299
  5. [5] B. DAHLBERG, On estimates of harmonic measure, Arch. Rational Mech. Anal., 65, N° 3 (1977), 275-288. Zbl0406.28009MR57 #6470
  6. [6] J. L. DOOB, A relativized Fatou theorem, Proc. Nat. Acad. Sc., 45 (1959), N° 2, 215-222. Zbl0106.07801MR21 #5822
  7. [7] K. GOWRISANKARAN, Fatou-Naim-Doob limit theorems in the axiomatic system of Brelot, Ann. Inst. Fourier, 16, 2 (1966), 455-467. Zbl0145.15103MR35 #1802
  8. [8] R. A. HUNT and R. L. WHEEDEN, On the boundary values of harmonic functions, Trans. Amer. Math. Soc., 132 (1968), 307-322. Zbl0159.40501MR37 #1634
  9. [9] R. A. HUNT and R. L. WHEEDEN, Positive harmonic functions on Lipschitz domains, Trans. Amer. Math. Soc., 147 (1970), 507-527. Zbl0193.39601MR43 #547
  10. [10] J. T. KEMPER, A boundary Harnack principle for Lipschitz domains and the principle of positive singularities, Comm. Pure Applied Math., 25 (1972), 247-255. Zbl0226.31007MR45 #2193
  11. [11] J.-M. WU, On functions subharmonic in a Lipschitz domain, Proc. Amer. Math. Soc. (to appear). Zbl0377.31007

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