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# Higher Regularity of the Free Boundary in a Semilinear System

Dec 2022

In this paper we are concerned with higher regularity properties of theelliptic system $\Delta\mathbf{u}=|\mathbf{u}|^{q-1}\mathbf{u}\chi_{\{|\mathbf{u}|>0\}},\qquad\mathbf{u}=(u^1,\dots,u^m)$ for $0\leq q<1$. We show analyticity of the regular part of the freeboundary $\partial\{|\mathbf{u}|>0\}$, analyticity of$|\mathbf{u}|^{\frac{1-q}2}$ and $\frac{\mathbf{u}}{|\mathbf{u}|}$ up to theregular part of the free boundary. Applying a variant of the partialhodograph-Legendre transformation and the implicit function theorem, we arriveat a degenerate equation, which introduces substantial challenges to be dealtwith. Along the lines of our study, we also establish a Cauchy-Kowalevski typestatement to show the local existence of solution when the free boundary andthe restriction of $\frac{\mathbf{u}}{|\mathbf{u}|}$ from both sides to thefree boundary are given as analytic data.

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