Boundary behaviour for p harmonic functions in Lipschitz and starlike Lipschitz ring domains

John L. Lewis; Kaj Nyström

Annales scientifiques de l'École Normale Supérieure (2007)

  • Volume: 40, Issue: 5, page 765-813
  • ISSN: 0012-9593

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Lewis, John L., and Nyström, Kaj. "Boundary behaviour for p harmonic functions in Lipschitz and starlike Lipschitz ring domains." Annales scientifiques de l'École Normale Supérieure 40.5 (2007): 765-813. <http://eudml.org/doc/82726>.

@article{Lewis2007,
author = {Lewis, John L., Nyström, Kaj},
journal = {Annales scientifiques de l'École Normale Supérieure},
language = {eng},
number = {5},
pages = {765-813},
publisher = {Elsevier},
title = {Boundary behaviour for p harmonic functions in Lipschitz and starlike Lipschitz ring domains},
url = {http://eudml.org/doc/82726},
volume = {40},
year = {2007},
}

TY - JOUR
AU - Lewis, John L.
AU - Nyström, Kaj
TI - Boundary behaviour for p harmonic functions in Lipschitz and starlike Lipschitz ring domains
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2007
PB - Elsevier
VL - 40
IS - 5
SP - 765
EP - 813
LA - eng
UR - http://eudml.org/doc/82726
ER -

References

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  1. [1] Aikawa H., Shanmugalingam N., Carleson type estimates for p harmonic functions and the conformal Martin boundary of John domains in metric measure spaces, Michigan Math. J.53 (1) (2005) 165-188. Zbl1076.31006MR2125540
  2. [2] Aikawa H., Kilpeläinen T., Shanmugalingam N., Zhong X., Boundary Harnack principle for p harmonic functions in smooth Euclidean domains, submitted for publication. Zbl1121.35060
  3. [3] Alt H.W., Caffarelli L., Existence and regularity for a minimum problem with free boundary, J. reine angew. Math.325 (1981) 105-144. Zbl0449.35105MR618549
  4. [4] Ancona A., Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien, Ann. Inst. Fourier (Grenoble)28 (4) (1978) 169-213. Zbl0377.31001MR513885
  5. [5] Bennewitz B., Lewis J., On the dimension of p harmonic measure, Ann. Acad. Sci. Fenn.30 (2005) 459-505. Zbl1194.35189MR2173375
  6. [6] Bennewitz B., Lewis J., On weak reverse Hölder inequalities for nondoubling harmonic measures, Complex Variables49 (7–9) (2004) 571-582. Zbl1068.31001
  7. [7] Caffarelli L., A Harnack inequality approach to the regularity of free boundaries. Part I. Lipschitz free boundaries are C 1 , α , Rev. Math. Iberoamericana3 (1987) 139-162. Zbl0676.35085MR990856
  8. [8] Caffarelli L., A Harnack inequality approach to the regularity of free boundaries. II. Flat free boundaries are Lipschitz, Comm. Pure Appl. Math.42 (1) (1989) 55-78. Zbl0676.35086MR973745
  9. [9] Caffarelli L., A Harnack inequality approach to the regularity of free boundaries. III. Existence theory, compactness, and dependence on X, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)15 (4) (1989) 583-602. Zbl0702.35249MR1029856
  10. [10] Caffarelli L., Fabes E., Mortola S., Salsa S., Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana J. Math.30 (4) (1981) 621-640. Zbl0512.35038MR620271
  11. [11] Coifmann R., Fefferman C., Weighted norm inequalities for maximal functions and singular integrals, Studia Math.51 (1974) 241-250. Zbl0291.44007MR358205
  12. [12] Dahlberg B., On estimates of harmonic measure, Arch. Ration. Mech. Anal.65 (1977) 275-288. Zbl0406.28009MR466593
  13. [13] DiBenedetto E., C 1 + α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal.7 (1983) 827-850. Zbl0539.35027MR709038
  14. [14] Eremenko A., Lewis J., Uniform limits of certain A-harmonic functions with applications to quasiregular mappings, Ann. Acad. Sci. Fenn. AI, Math.16 (1991) 361-375. Zbl0727.35022MR1139803
  15. [15] Fabes E., Kenig C., Serapioni R., The local regularity of solutions to degenerate elliptic equations, Comm. Partial Differential Equations7 (1) (1982) 77-116. Zbl0498.35042MR643158
  16. [16] Fabes E., Jerison D., Kenig C., The Wiener test for degenerate elliptic equations, Ann. Inst. Fourier (Grenoble)32 (3) (1982) 151-182. Zbl0488.35034MR688024
  17. [17] Fabes E., Jerison D., Kenig C., Boundary behavior of solutions to degenerate elliptic equations, in: Conference on Harmonic Analysis in Honor of Antonio Zygmund, vols. I, II, Chicago, IL, 1981, Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 577-589. Zbl0503.35038MR730093
  18. [18] Gehring F., On the L p integrability of the derivatives of a quasiconformal mapping, Acta Math.130 (1973) 265-277. Zbl0258.30021MR402038
  19. [19] Gariepy R., Ziemer W.P., A regularity condition at the boundary for solutions of quasilinear elliptic equations, Arch. Ration. Mech. Anal.67 (1977) 25-39. Zbl0389.35023MR492836
  20. [20] Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order, second ed., Springer-Verlag, 1983. Zbl0562.35001MR737190
  21. [21] Heinonen J., Kilpeläinen T., Martio O., Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, 1993. Zbl0780.31001MR1207810
  22. [22] Hofmann S., Lewis J., The Dirichlet problem for parabolic operators with singular drift term, Mem. Amer. Math. Soc.151 (719) (2001) 1-113. Zbl1149.35048MR1828387
  23. [23] Jerison D., Regularity of the Poisson kernel and free boundary problems, Colloq. Math.60–61 (1990) 547-567. Zbl0732.35025
  24. [24] Jerison D., Kenig C., Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. Math.46 (1982) 80-147. Zbl0514.31003MR676988
  25. [25] Jerison D., Kenig C., The logarithm of the Poisson kernel of a C 1 domain has vanishing mean oscillation, Trans. Amer. Math. Soc.273 (1984) 781-794. Zbl0494.31003MR667174
  26. [26] Kemper J., A boundary Harnack inequality for Lipschitz domains and the principle of positive singularities, Comm. Pure Appl. Math.25 (1972) 247-255. Zbl0226.31007MR293114
  27. [27] Kenig C., Toro T., Harmonic measure on locally flat domains, Duke Math. J.87 (1997) 501-551. Zbl0878.31002MR1446617
  28. [28] Kenig C., Toro T., Free boundary regularity for harmonic measure and Poisson kernels, Ann. of Math.150 (1999) 369-454. Zbl0946.31001MR1726699
  29. [29] Kenig C., Toro T., Poisson kernel characterization of Reifenberg flat chord arc domains, Ann. Sci. Ecole Norm. Sup. (4)36 (3) (2003) 323-401. Zbl1027.31005MR1977823
  30. [30] Kenig C., Toro T., Free boundary regularity below the continuous threshold: 2-phase problems, submitted for publication. Zbl1106.35147
  31. [31] Kenig C., Pipher J., The Dirichlet problem for elliptic operators with drift term, Publ. Mat.45 (1) (2001) 199-217. Zbl1113.35314MR1829584
  32. [32] Kilpeläinen T., Zhong X., Growth of entire A-subharmonic functions, Ann. Acad. Sci. Fenn. AI, Math.28 (2003) 181-192. Zbl1018.35027MR1976839
  33. [33] Krol' I.N., On the behavior of the solutions of a quasilinear equation near null salient points of the boundary, Proc. Steklov Inst. Math.125 (1973) 130-136. Zbl0306.35047MR344671
  34. [34] Lewis J., Vogel A., Uniqueness in a free boundary problem, Comm. Partial Differential Equations31 (2006) 1591-1614. Zbl1189.35130MR2273966
  35. [35] Lewis J., Vogel A., Symmetry problems and uniform rectifiability, submitted for publication. 
  36. [36] Lewis J., Note on p harmonic measure, Comput. Methods Funct. Theory6 (1) (2006) 109-144. Zbl1161.35406MR2241036
  37. [37] Lewis J., Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. J.32 (6) (1983) 849-858. Zbl0554.35048MR721568
  38. [38] Lieberman G.M., Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal.12 (11) (1988) 1203-1219. Zbl0675.35042MR969499
  39. [39] Littman W., Stampacchia G., Weinberger H.F., Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa (3)17 (1963) 43-77. Zbl0116.30302MR161019
  40. [40] Mattila P., Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, 1995. Zbl0819.28004MR1333890
  41. [41] Serrin J., Local behavior of solutions of quasilinear elliptic equations, Acta Math.111 (1964) 247-302. Zbl0128.09101MR170096
  42. [42] Stein E.M., Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970. Zbl0207.13501MR290095
  43. [43] Tolksdorff P., Everywhere regularity for some quasi-linear systems with lack of ellipticity, Ann. Math. Pura Appl.134 (4) (1984) 241-266. Zbl0538.35034MR736742
  44. [44] Wu J.M., Comparisons of kernel functions, boundary Harnack principle and relative Fatou theorem on Lipschitz domains, Ann. Inst. Fourier (Grenoble)28 (4) (1978) 147-167. Zbl0368.31006MR513884

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