# American Institute of Mathematical Sciences

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

## On the free boundary regularity theorem of Alt and Caffarelli

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