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2018 Quantitative stability of the free boundary in the obstacle problem
Sylvia Serfaty, Joaquim Serra
Anal. PDE 11(7): 1803-1839 (2018). DOI: 10.2140/apde.2018.11.1803

Abstract

We prove some detailed quantitative stability results for the contact set and the solution of the classical obstacle problem in 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 ] × 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.

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Sylvia Serfaty. Joaquim Serra. "Quantitative stability of the free boundary in the obstacle problem." Anal. PDE 11 (7) 1803 - 1839, 2018. https://doi.org/10.2140/apde.2018.11.1803

Information

Received: 10 August 2017; Revised: 13 February 2018; Accepted: 9 April 2018; Published: 2018
First available in Project Euclid: 15 January 2019

zbMATH: 1391.35432
MathSciNet: MR3810473
Digital Object Identifier: 10.2140/apde.2018.11.1803

Subjects:
Primary: 31B35, 35R35, 49K99

Rights: Copyright © 2018 Mathematical Sciences Publishers

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Vol.11 • No. 7 • 2018
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