Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

On the free boundary regularity theorem of Alt and Caffarelli

Pages: 397 - 422, Volume 10, Issue 1/2, January/February 2004      doi:10.3934/dcds.2004.10.397

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Carlos E. Kenig - Department of Mathematics, University of Chicago, Chicago, Illinois, 60637–1514, United States (email)
Tatiana Toro - Department of Mathematics, University of Washington, Seattle, Washington 98195–4350, 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.
Mathematics Subject Classification:  34A26.

Received: May 2002;      Revised: April 2003;      Available Online: October 2003.