Quantitative stability of the free boundary in the obstacle problem

Sylvia Serfaty, Joaquim Serra

Research output: Contribution to journalArticlepeer-review

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 languageEnglish (US)
Pages (from-to)1803-1839
Number of pages37
JournalAnalysis and PDE
Volume11
Issue number7
DOIs
StatePublished - 2018

Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Applied Mathematics

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