Abstract
We prove some detailed quantitative stability results for the contact set and the solution of the classical obstacle problem in ℝn (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 Γt corresponding to the boundary of the contact set for a family of obstacle functions ht. Assuming that h = ht(x) = h(t, x) is Ck+1, α in [-1, 1]×ℝn and that the initial free boundary Γ0 is regular, we prove that Γt is twice differentiable in t in a small neighborhood of t = 0. Moreover, we show that the "normal velocity" and the "normal acceleration" of Γt are respectively Ck-1, α and Ck-2, α scalar fields on Γt. 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) |
---|---|
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