Abstract

We consider an obstacle-type problem Δu = f(x)χΩ     in D, u = |∇u| = 0     on D \ Ω, where D is a given open set in ℝn and Ω is an unknown open subset of D. The problem originates in potential theory, in connection with harmonic continuation of potentials. The qualitative difference between this problem and the classical obstacle problem is that the solutions here are allowed to change sign. Using geometric and energetic criteria in delicate combination we show the C1,1 regularity of the solutions, and the regularity of the free boundary, below the Lipschitz threshold for the right-hand side.

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Additional Information

ISSN
1080-6377
Print ISSN
0002-9327
Pages
pp. 1659-1688
Launched on MUSE
2007-12-10
Open Access
No
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