Boundary behaviour for $p$ harmonic functions in Lipschitz and starlike Lipschitz ring domains
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 40 (2007) no. 5, pp. 765-813.
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author = {Lewis, John L. and Nystr\"om, Kaj},
title = {Boundary behaviour for $p$ harmonic functions in {Lipschitz} and starlike {Lipschitz} ring domains},
journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
pages = {765--813},
publisher = {Elsevier},
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Lewis, John L.; Nyström, Kaj. Boundary behaviour for $p$ harmonic functions in Lipschitz and starlike Lipschitz ring domains. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 40 (2007) no. 5, pp. 765-813. doi : 10.1016/j.ansens.2007.09.001. http://www.numdam.org/articles/10.1016/j.ansens.2007.09.001/

[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. | MR | Zbl

[2] Aikawa H., Kilpeläinen T., Shanmugalingam N., Zhong X., Boundary Harnack principle for p harmonic functions in smooth Euclidean domains, submitted for publication. | Zbl

[3] Alt H.W., Caffarelli L., Existence and regularity for a minimum problem with free boundary, J. reine angew. Math. 325 (1981) 105-144. | MR | Zbl

[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. | Numdam | MR | Zbl

[5] Bennewitz B., Lewis J., On the dimension of p harmonic measure, Ann. Acad. Sci. Fenn. 30 (2005) 459-505. | MR

[6] Bennewitz B., Lewis J., On weak reverse Hölder inequalities for nondoubling harmonic measures, Complex Variables 49 (7-9) (2004) 571-582. | Zbl

[7] Caffarelli L., A Harnack inequality approach to the regularity of free boundaries. Part I. Lipschitz free boundaries are ${C}^{1,\alpha }$, Rev. Math. Iberoamericana 3 (1987) 139-162. | MR | Zbl

[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. | MR | Zbl

[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. | Numdam | MR

[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. | MR | Zbl

[11] Coifmann R., Fefferman C., Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974) 241-250. | MR | Zbl

[12] Dahlberg B., On estimates of harmonic measure, Arch. Ration. Mech. Anal. 65 (1977) 275-288. | MR | Zbl

[13] Dibenedetto E., ${C}^{1+\alpha }$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983) 827-850. | MR | Zbl

[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. | MR | Zbl

[15] Fabes E., Kenig C., Serapioni R., The local regularity of solutions to degenerate elliptic equations, Comm. Partial Differential Equations 7 (1) (1982) 77-116. | MR | Zbl

[16] Fabes E., Jerison D., Kenig C., The Wiener test for degenerate elliptic equations, Ann. Inst. Fourier (Grenoble) 32 (3) (1982) 151-182. | Numdam | MR | Zbl

[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. | MR | Zbl

[18] Gehring F., On the ${L}^{p}$ integrability of the derivatives of a quasiconformal mapping, Acta Math. 130 (1973) 265-277. | MR | Zbl

[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. | MR | Zbl

[20] Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order, second ed., Springer-Verlag, 1983. | MR | Zbl

[21] Heinonen J., Kilpeläinen T., Martio O., Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, 1993. | MR | Zbl

[22] Hofmann S., Lewis J., The Dirichlet problem for parabolic operators with singular drift term, Mem. Amer. Math. Soc. 151 (719) (2001) 1-113. | MR

[23] Jerison D., Regularity of the Poisson kernel and free boundary problems, Colloq. Math. 60-61 (1990) 547-567. | Zbl

[24] Jerison D., Kenig C., Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. Math. 46 (1982) 80-147. | MR | Zbl

[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. | MR | Zbl

[26] Kemper J., A boundary Harnack inequality for Lipschitz domains and the principle of positive singularities, Comm. Pure Appl. Math. 25 (1972) 247-255. | MR | Zbl

[27] Kenig C., Toro T., Harmonic measure on locally flat domains, Duke Math. J. 87 (1997) 501-551. | MR | Zbl

[28] Kenig C., Toro T., Free boundary regularity for harmonic measure and Poisson kernels, Ann. of Math. 150 (1999) 369-454. | MR | Zbl

[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. | Numdam | MR | Zbl

[30] Kenig C., Toro T., Free boundary regularity below the continuous threshold: 2-phase problems, submitted for publication. | Zbl

[31] Kenig C., Pipher J., The Dirichlet problem for elliptic operators with drift term, Publ. Mat. 45 (1) (2001) 199-217. | MR | Zbl

[32] Kilpeläinen T., Zhong X., Growth of entire A-subharmonic functions, Ann. Acad. Sci. Fenn. AI, Math. 28 (2003) 181-192. | MR | Zbl

[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. | MR | Zbl

[34] Lewis J., Vogel A., Uniqueness in a free boundary problem, Comm. Partial Differential Equations 31 (2006) 1591-1614. | MR

[35] Lewis J., Vogel A., Symmetry problems and uniform rectifiability, submitted for publication.

[36] Lewis J., Note on p harmonic measure, Comput. Methods Funct. Theory 6 (1) (2006) 109-144. | MR

[37] Lewis J., Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. J. 32 (6) (1983) 849-858. | MR | Zbl

[38] Lieberman G.M., Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (11) (1988) 1203-1219. | MR | Zbl

[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. | Numdam | MR | Zbl

[40] Mattila P., Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, 1995. | MR | Zbl

[41] Serrin J., Local behavior of solutions of quasilinear elliptic equations, Acta Math. 111 (1964) 247-302. | MR | Zbl

[42] Stein E.M., Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970. | MR | Zbl

[43] Tolksdorff P., Everywhere regularity for some quasi-linear systems with lack of ellipticity, Ann. Math. Pura Appl. 134 (4) (1984) 241-266. | MR | Zbl

[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. | Numdam | MR | Zbl

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