Quantitative stability of the free boundary in the obstacle problem
Quantitative stability of the free boundary in the obstacle problem
复制标题
障碍问题中自由边界的定量稳定性
DOI：
10.2140/apde.2018.11.1803
复制
发表时间：
2017
期刊：
影响因子：
--
通讯作者：
J. Serra
中科院分区：
文献类型：
--
作者：
S. Serfaty;J. Serra
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.
要做到这一点，在整个空间的设置工作，我们检查的自由边界$\Gamma^t$对应的接触集的边界的障碍函数的家庭$h^t$的演变。假设$h=h^t（x）= h（t，x）$是$C^{k+1，\alpha}$在$[-1，1]\times \mathbb{R}^n$中，且初始自由边界$\Gamma ^0 $是正则的，我们证明了$\Gamma^t$在$t$中在$t=0$的一个小邻域内是二次可微的.此外，我们还证明了$\Gamma^t$的“法向速度”和“法向加速度”分别是$\Gamma ^t$上的$C^{k-1，\alpha}$和$C^{k-2，\alpha}$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。
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 $\Gamma^t$ 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 $C^{k+1,\alpha}$ in $[-1,1]\times \mathbb{R}^n$ and that the initial free boundary $\Gamma^0$ is regular, we prove that $\Gamma^t$ is twice differentiable in $t$ in a small neighborhood of $t=0$. Moreover, we show that the "normal velocity" and the "normal acceleration" of $\Gamma^t$ are respectively $C^{k-1,\alpha}$ and $C^{k-2,\alpha}$ scalar fields on $\Gamma^t$. 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.