A boundary Harnack principle in twisted Hölder domains. (English) Zbl 0747.31008

The boundary Harnack principle for the ratio of positive harmonic functions is shown to hold in Hölder domains and twisted Hölder domains, respectively, of order \(\alpha\), \(\alpha\in]1/2,1]\). For Lipschitz domains, i.e. in the case \(\alpha=1\), this was proved by A. Ancona [Ann. Inst. Fourier 28, 169-213 (1978; Zbl 0386.31002)] and by B. Dahlberg [Arch. Rat. Mech. Anal. 65, 275-288, (1977; Zbl 0406.31003)]. The author’s results are proved using the connection between Brownian motion and harmonic functions. It is shown that the desired boundary Harnack principle holds also for \(L\)-harmonic functions for a uniformly elliptic operator \(L\) in divergence form.
Further, a twisted Hölder domain of order \(\alpha\in]0,1/2[\) is constructed to show that in this case the boundary Harnack principle is in general invalid.


31C35 Martin boundary theory
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
60J45 Probabilistic potential theory
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