On the free boundary regularity theorem of Alt and Caffarelli
Carlos E. Kenig - Department of Mathematics, University of Chicago, Chicago, Illinois, 60637–1514, United States (email)
Abstract: In this note we discuss a slight generalization of the following result by Alt and Caffarelli: if the logarithm of the Poisson kernel of a Reifenberg flat chord arc domain is Hölder continuous, then the domain can be locally represented as the area above the graph of a function whose gradient is Hölder continuous. In this note we show that if the Poisson kernel of an unbounded Reifenberg flat chord arc domain is 1 a.e. on the boundary then the domain is (modulo rotation and translation) the upper half plane. This result plays a key role in the study of regularity of the free boundary below the continuous threshold.
Keywords: Reifenberg ﬂat chord arc domain, Poisson kernel.
Received: May 2002; Revised: April 2003; Published: October 2003.
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