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Boundary behaviour of \(p\)-harmonic functions in domains beyond Lipschitz domains. (English) Zbl 1169.31004

The authors prove the boundary Harnack inequality and Hölder continuity for ratios of \(p\)-harmonic functions vanishing on a portion of certain Reifenberg flat and Ahlfors regular NTA-domains, which extends a number of results on the boundary behavior of \(p\)-harmonic functions. Precisely, in a series of the previous papers, the authors proved a number of results concerning the boundary behavior of positive \(p\)-harmonic functions, \(1 < p < \infty\), in a bounded Lipschitz domain. In particular, they established the boundary Harnack inequality as well as Hölder continuity for ratios of positive \(p\)-harmonic functions, \(1 < p < \infty\), vanishing on a portion of the boundary.
In the present paper, the authors consider certain Reifenberg flat with vanishing constant and Ahlfors regular NTA-domains, beyond Lipschitz domains. Neither class of domains is contained in the other and their proofs of the \(p\)-harmonic boundary Harnack results differ at key points for each class. But these results cannot to be extended to the general setting of NTA-domains for \(1 < p < \infty, p \neq 2\). In addition, the Martin boundary problem is also considered in these classes of domains.

MSC:

31C45 Other generalizations (nonlinear potential theory, etc.)
31C35 Martin boundary theory
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[1] Ancona A., Ann. Inst. Fourier 28 pp 4– (1978)
[2] Bennewitz B., Ann. Acad. Sci. Fenn. 30 pp 459– (2005)
[3] Dahlberg B., Arch Rational Mech. Anal. 65 pp 275– (1977) · Zbl 0406.28009 · doi:10.1007/BF00280445
[4] David G., Indiana Univ. Math. J. 39 pp 3– (1990)
[5] Fabes E., Ann. Inst. Fourier (Grenoble) 32 pp 3– (1982)
[6] Fabes E., Ill. pp 577– (1981)
[7] Fabes E., Comm. Partial Differential Equations 7 pp 1– (1982) · Zbl 0525.35064 · doi:10.1080/03605308208820216
[8] DOI: 10.1016/0001-8708(82)90055-X · Zbl 0514.31003 · doi:10.1016/0001-8708(82)90055-X
[9] Kemper J., Comm. Pure Appl. Math. 25 pp 247– (1972) · Zbl 0226.31007 · doi:10.1002/cpa.3160250303
[10] Kenig C., Publ. Mat. 45 pp 1– (2001)
[11] Kenig C., Duke Math J. 87 pp 501– (1997)
[12] Lewis J., Computational Methods in Function Theory 6 pp 1– (2006)
[13] Lewis J., Sup. 40 pp 765– (4)
[14] Lewis J., Ann. Acad. Sci. Fenn. 33 pp 1– (2008)
[15] Littman W., Ann. Scuola Norm. Sup. Pisa 17 pp 43– (3)
[16] Wu J. M., Ann. Inst. Fourier (Grenoble) 28 pp 4– (1978)
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