Regularity Properties of a Free Boundary near Contact Points with the Fixed Boundary

In the upper half of the unit ball B = {|x | < 1, x1 > 0}, let u and  (a domain in R+ = {x ∈ R n : x1 > 0} ) solve the following overdetermined problem: 1u = χ in B , u = |∇u| = 0 in B \, u = 0 on 5 ∩ B, where B is the unit ball with center at the origin, χ denotes the characteristic function of , 5 = {x1 = 0}, n ≥ 2, and the equation is satisfied in the sense of distributions. We show (among other things) that if the origin is a contact point of the free boundary, that is, if u(0) = |∇u(0)| = 0, then ∂∩ Br0 is the graph of a C 1-function over 5 ∩ Br0 . The C 1-norm depends on the dimension and supB+ |u|. The result is extended to general subdomains of the unit ball with C3-boundary.