We provide a higher order boundary Harnack inequality for harmonic functions in slit domains. As a corollary we obtain the regularity of the free boundary in the Signorini problem near non-degenerate points.
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Daniela De Silva, Ovidiu Savin, Boundary Harnack estimates in slit domains and applications to thin free boundary problems. Rev. Mat. Iberoam. 32 (2016), no. 3, pp. 891–912DOI 10.4171/RMI/902