两相膜问题--求分支点正则性的交叉比较法
The Two-Phase Membrane Problem -- an Intersection-Comparison Approach to the Regularity at Branch Points
摘要：
对于两相膜问题$ Δu = {λ_+\over 2} χ_{\{u>0\}} - {λ_-\over 2} χ_{\{u<0\}} ,$，其中$λ_+> 0$和$λ_->0 ,$，我们在二维证明了自由边界在两个$C^1$ -图的并的每个"分支点“的邻域内.我们还得到了稳定性结果的扰动的边界数据。我们的分析使用的交叉比较的方法的基础上亚历山德罗夫反射。在更高的维度，我们表明，自由边界有有限的$(n-1)$维Hausdorff措施。
英文摘要：
For the two-phase membrane problem $ Δu = {λ_+\over 2} χ_{\{u>0\}} - {λ_-\over 2} χ_{\{u<0\}} ,$ where $λ_+> 0$ and $λ_->0 ,$ we prove in two dimensions that the free boundary is in a neighborhood of each ``branch point'' the union of two $C^1$-graphs. We also obtain a stability result with respect to perturbations of the boundary data. Our analysis uses an intersection-comparison approach based on the Aleksandrov reflection. In higher dimensions we show that the free boundary has finite $(n-1)$-dimensional Hausdorff measure.
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