On the free boundary regularity theorem of Alt and Caffarelli

Carlos E. Kenig; Tatiana Toro

doi:10.3934/dcds.2004.10.397

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2004, Volume 10, Issue 1&2: 397-422. Doi: 10.3934/dcds.2004.10.397

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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
2.
Department of Mathematics, University of Washington, Seattle, Washington 98195–4350

Received: May 2002

Revised: April 2003

Published: January 2004

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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.

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