论文标题
半线性系统中自由边界的较高规律性
Higher Regularity of the Free Boundary in a Semilinear System
论文作者
论文摘要
在本文中，我们关注椭圆系统\ [δ\ mathbf {u} =的较高规律性属性| \ Mathbf {我们显示了自由边界$ \ partial \ {| \ MathBf {u} |> 0 \} $的分析性，$ | \ Mathbf {u} |^{u} |^{\ frac {1-q} 2} 2} 2} $ and and $ \ frac {\ mathbf {\ mathbf {\ to {自由边界的一部分。采用部分Hodograph-legendre变换和隐式函数定理的变体，我们得出了一个退化方程，这引入了要解决的重大挑战。 按照我们的研究界限，我们还建立了一个cauchy-kowalevski类型语句，以显示自由边界的局部存在以及$ \ frac {\ mathbf {u}} {\ mathbf {u}} {| \ mathbf {U} |} $从双方从双方提供的分析数据。
In this paper we are concerned with higher regularity properties of the elliptic system \[ Δ\mathbf{u}= |\mathbf{u}|^{q-1}\mathbf{u}χ_{\{|\mathbf{u}|>0\}},\qquad\mathbf{u}=(u^1,\dots,u^m) \] for $0\leq q<1$. We show analyticity of the regular part of the free boundary $\partial\{|\mathbf{u}|>0\}$, analyticity of $|\mathbf{u}|^{\frac{1-q}2} $ and $ \frac{\mathbf{u}}{|\mathbf{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 $ \frac{\mathbf{u}}{|\mathbf{u}|} $ from both sides to the free boundary are given as analytic data.
