论文标题
堕落的一阶段自由结合问题
One-phase free-boundary problems with degeneracy
论文作者
论文摘要
在本文中，我们研究了Alt-Caffarelli功能的退化版本的本地最小化器。具体而言，我们考虑功能$ j_ {q}（u，ω）的本地最小化器：= \int_Ω| \ nabla u |^2 + q（x）^2χ_ {\ {\ {u> 0 \}} dx $ were $ q（x） α} $ submanifold of Dimension $ 0 \ le K \ le n-1 $。我们表明，自由边界可以分解为可重新分配的集合，我们证明了Minkowski含量估计值，而退化的尖齿套件通常可以用当前技术来说几乎没有什么。水波理论和斯托克斯波的作品是我们的灵感，但是本文的主要目的是研究在局部极简化器的背景下，自由边界的自由边界的几何形状。
In this paper, we study local minimizers of a degenerate version of the Alt-Caffarelli functional. Specifically, we consider local minimizers of the functional $J_{Q}(u, Ω):= \int_Ω |\nabla u|^2 + Q(x)^2χ_{\{u>0\}}dx$ where $Q(x) = \text{dist}(x, Γ)^γ$ for $γ>0$ and $Γ$ a $C^{1, α}$ submanifold of dimension $0 \le k \le n-1$. We show that the free boundary may be decomposed into a rectifiable set, on which we prove upper Minkowski content estimates, and a degenerate cusp set about which little can be said in general with the current techniques. Work in the theory of water waves and the Stokes wave serves as our inspiration, however the main thrust of this paper is to study the geometry of the free boundary for degenerate one-phase Bernoulli free-boundary problems in the context of local minimizers.
