Higher regularity of the free boundary in a semilinear system

In this paper we are concerned with higher regularity properties of the elliptic system (Formula presented.) for 0≤q<1. We show analyticity of the regular part of the free boundary ∂{|u|>0}, analyticity of |u|^1-q2 and u|u| up to the regular part of the free boundary. Applying a variant of the partial hodograph-Legendre transformation and the implicit function theorem, we arrive at a degenerate equation, which introduces substantial challenges to be dealt with. Along the lines of our study, we also establish a Cauchy-Kowalevski type statement to show the local existence of solution when the free boundary and the restriction of u|u| from both sides to the free boundary are given as analytic data.