Boundary Regularity and Compactness for Overdetermined Problems

1 Let D be either the unit ball B1(0) or the half ball B + 1 (0), let f be a strictly positive and continuous function, and let u and Ω ⊂ D solve the following overdetermined problem: ∆u(x) = χ Ω (x)f(x) in D, 0 ∈ ∂Ω, u = |∇u| = 0 in Ω, where χ Ω denotes the characteristic function of Ω, Ωc denotes the set D \Ω, and the equation is satisfied in the sense of distributions. When D = B 1 (0), then we impose in addition that u(x) ≡ 0 on { (x, xn) | xn = 0 } . We show that a fairly mild thickness assumption on Ωc will ensure enough compactness on u to give us “blow-up” limits, and we show how this compactness leads to regularity of ∂Ω. In the case where f is positive and Lipschitz, the methods developed in Caffarelli, Karp, and Shahgholian (2000) lead to regularity of ∂Ω under a weaker thickness assumption.