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- W2744164817 startingPage "1803" @default.
- W2744164817 abstract "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." @default.
- W2744164817 created "2017-08-17" @default.
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- W2744164817 date "2018-05-20" @default.
- W2744164817 modified "2023-10-18" @default.
- W2744164817 title "Quantitative stability of the free boundary in the obstacle problem" @default.
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- W2744164817 doi "https://doi.org/10.2140/apde.2018.11.1803" @default.
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