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2021 On the singular set in the thin obstacle problem: higher-order blow-ups and the very thin obstacle problem
Xavier Fernández-Real, Yash Jhaveri
Anal. PDE 14(5): 1599-1669 (2021). DOI: 10.2140/apde.2021.14.1599

Abstract

We consider the singular set in the thin obstacle problem with weight |xn+1|a for a(1,1), which arises as the local extension of the obstacle problem for the fractional Laplacian (a nonlocal problem). We develop a refined expansion of the solution around its singular points by building on the ideas introduced by Figalli and Serra to study the fine properties of the singular set in the classical obstacle problem. As a result, under a superharmonicity condition on the obstacle, we prove that each stratum of the singular set is locally contained in a single C2 manifold, up to a lower-dimensional subset, and the top stratum is locally contained in a C1,α manifold for some α>0 if a<0.

In studying the top stratum, we discover a dichotomy, until now unseen, in this problem (or, equivalently, the fractional obstacle problem). We find that second blow-ups at singular points in the top stratum are global, homogeneous solutions to a codimension-2 lower-dimensional obstacle problem (or fractional thin obstacle problem) when a<0, whereas second blow-ups at singular points in the top stratum are global, homogeneous, and a-harmonic polynomials when a0. To do so, we establish regularity results for this codimension-2 problem, which we call the very thin obstacle problem.

Our methods extend to the majority of the singular set even when no sign assumption on the Laplacian of the obstacle is made. In this general case, we are able to prove that the singular set can be covered by countably many C2 manifolds, up to a lower-dimensional subset.

Citation

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Xavier Fernández-Real. Yash Jhaveri. "On the singular set in the thin obstacle problem: higher-order blow-ups and the very thin obstacle problem." Anal. PDE 14 (5) 1599 - 1669, 2021. https://doi.org/10.2140/apde.2021.14.1599

Information

Received: 15 October 2019; Accepted: 5 February 2020; Published: 2021
First available in Project Euclid: 6 January 2022

Digital Object Identifier: 10.2140/apde.2021.14.1599

Subjects:
Primary: 35R35 , 47G20

Keywords: fractional Laplacian , free boundary , obstacle problem

Rights: Copyright © 2021 Mathematical Sciences Publishers

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Vol.14 • No. 5 • 2021
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