Regularity of a Free Boundary with Application to the Pompeiu Problem

In the unit ball B(0, 1), let u and Ω (a domain in RN) solve the following overdetermined problem: Δ u = χΩin B(0, 1), 0 ∈ ∂ Ω , u = |Δ u| = 0 in B(0, 1)$\backslash \ \Omega $, where χΩdenotes the characteristic function, and the equation is satisfied in the sense of distributions. If the complement...

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Bibliographic Details
Published in:Annals of mathematics 2000-01, Vol.151 (1), p.269-292
Main Authors: Caffarelli, Luis A., Karp, Lavi, Shahgholian, Henrik
Format: Article
Language:English
Subjects:
dimensions
growth
Infinity
Lebesgue measures
Mathematical functions
Mathematical inequalities
Mathematical monotonicity
Mathematical theorems
Maximum principle
Polynomials
quadrature domains
spheres
Unit ball
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Summary:In the unit ball B(0, 1), let u and Ω (a domain in RN) solve the following overdetermined problem: Δ u = χΩin B(0, 1), 0 ∈ ∂ Ω , u = |Δ u| = 0 in B(0, 1)$\backslash \ \Omega $, where χΩdenotes the characteristic function, and the equation is satisfied in the sense of distributions. If the complement of Ω does not develop cusp singularities at the origin then we prove ∂ Ω is analytic in some small neighborhood of the origin. The result can be modified to yield for more general divergence form operators. As an application of this, then, we obtain the regularity of the boundary of a domain without the Pompeiu property, provided its complement has no cusp singularities.
ISSN:0003-486X
1939-8980
DOI:10.2307/121117