## Abstract

We prove some detailed quantitative stability results for the contact set and the solution of the classical obstacle problem in ℝ^{n} (n ≥ 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]×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 this velocity and acceleration and studying the regularity of their solutions via single- and double-layer estimates from potential theory.

Original language | English (US) |
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Pages (from-to) | 1803-1839 |

Number of pages | 37 |

Journal | Analysis and PDE |

Volume | 11 |

Issue number | 7 |

DOIs | |

State | Published - 2018 |

## Keywords

- Coincidence set
- Contact set
- Equilibrium measure
- Obstacle problem
- Potential theory
- Stability

## ASJC Scopus subject areas

- Analysis
- Numerical Analysis
- Applied Mathematics