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arXiv 1708.01490math.APmath-phmath.CVmath.MP
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障碍问题中自由边界的定量稳定性

Quantitative stability of the free boundary in the obstacle problem

Sylvia Serfaty, Joaquim Serra

备注：
37 pages
发表信息：
Analysis & PDE 11 (2018) 1803-1839
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摘要：

我们证明了在障碍函数扰动下$\mathbb{R}^n$（$n \ge 2$）中接触集和经典障碍问题解的一些详细的定量稳定性结果，这也等价于研究经典位势理论中平衡测度在外场扰动下的变化.为此，在整个空间的设置工作，我们研究的自由边界$Γ^t$对应的接触集的边界的障碍函数$h^t$家庭的演变。假设$h=h^t (x) = h(t,x)$是$[-1,1]\times \mathbb{R}^n$中的$C^{k+1,α}$，且初始自由边界$Γ^0$是正则的，我们证明了在$t=0$的一个小邻域中，$Γ^t$在$t$中是二次可微的.进一步证明了$Γ^t$的“法向速度”和“法向加速度”分别是$C^{k-1,α}$和$C^{k-2,α}$在$Γ^t$上的标量场.这是通过推导这些速度和加速度的方程，并研究其解决方案的规律性，通过单，双层估计从潜在的理论。

英文摘要：

We prove some detailed quantitative stability results for the contact set and the solution of the classical obstacle problem in $\mathbb{R}^n$ ($n \ge 2$) under perturbations of the obstacle function, which is also equivalent to studying the variation of the equilibrium measure in classical potential theory under a perturbation of the external field. To do so, working in the setting of the whole space, we examine the evolution of the free boundary $Γ^t$ corresponding to the boundary of the contact set for a family of obstacle functions $h^t$. Assuming that $h=h^t (x) = h(t,x)$ is $C^{k+1,α}$ in $[-1,1]\times \mathbb{R}^n$ and that the initial free boundary $Γ^0$ is regular, we prove that $Γ^t$ is twice differentiable in $t$ in a small neighborhood of $t=0$. Moreover, we show that the "normal velocity" and the "normal acceleration" of $Γ^t$ are respectively $C^{k-1,α}$ and $C^{k-2,α}$ scalar fields on $Γ^t$. This is accomplished by deriving equations for these velocity and acceleration and studying the regularity of their solutions via single and double layers estimates from potential theory.

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