On the singular set in the thin obstacle problem: higher-order blow-ups and the very thin obstacle problem

Fernández-Real, Xavier; Jhaveri, Yash

doi:10.2140/apde.2021.14.1599

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On the singular set in the thin obstacle problem: higher-order blow-ups and the very thin obstacle problem

Xavier Fernández-Real and Yash Jhaveri

Vol. 14 (2021), No. 5, 1599–1669

DOI: 10.2140/apde.2021.14.1599

Abstract

We consider the singular set in the thin obstacle problem with weight $| x_{n + 1} |^{a}$ for $a \in (- 1, 1)$ , 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 $C^{2}$ manifold, up to a lower-dimensional subset, and the top stratum is locally contained in a $C^{1, α}$ 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 $a \geq 0$ . 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 $C^{2}$ manifolds, up to a lower-dimensional subset.

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Keywords

obstacle problem, fractional Laplacian, free boundary

Mathematical Subject Classification 2010

Primary: 35R35, 47G20

Milestones

Received: 15 October 2019

Accepted: 5 February 2020

Published: 22 August 2021

Authors

Xavier Fernández-Real
Institute of Mathematics EPFL Lausanne Switzerland

Yash Jhaveri
Institute for Advanced Study Princeton, NJ United States

Vol. 14, No. 5, 2021