Higher Regularity of the Free Boundary in a Semilinear System

In this paper we are concerned with higher regularity properties of the elliptic system $$\mathrm{\Delta }\mathbf{u}=\mathrm{\mid }\mathbf{u}{\mathrm{\mid }}^{q-1}\mathbf{u}{\chi }_{\{\mathrm{\mid }\mathbf{u}\mathrm{\mid }>0\}},\mathbf{u}=(u^1,\dots ,u^m)$$ for $0\le q<1$. We show analyticity of the regular part of the free boundary $\mathrm{\partial }\{\mathrm{\mid }\mathbf{u}\mathrm{\mid }>0\}$, analyticity of $\mathrm{\mid }\mathbf{u}{\mathrm{\mid }}^{\frac{1-q}{2}}$ and $\frac{\mathbf{u}}{\mathrm{\mid }\mathbf{u}\mathrm{\mid }}$ 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 $\frac{\mathbf{u}}{\mathrm{\mid }\mathbf{u}\mathrm{\mid }}$ from both sides to the free boundary are given as analytic data.