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February  2004, 10(1&2): 397-422. doi: 10.3934/dcds.2004.10.397

On the free boundary regularity theorem of Alt and Caffarelli

Carlos E. Kenig 1, and  Tatiana Toro 2,

1. 

Department of Mathematics, University of Chicago, Chicago, Illinois, 60637–1514, United States
2. 

Department of Mathematics, University of Washington, Seattle, Washington 98195–4350, United States

Received  May 2002 Revised  April 2003 Published  October 2003

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: 34A2.
Citation: Carlos E. Kenig, Tatiana Toro. On the free boundary regularity theorem of Alt and Caffarelli. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 397-422. doi: 10.3934/dcds.2004.10.397
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