Vol. 14, No. 5, 2021

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 4, 1127–1500
Issue 3, 757–1126
Issue 2, 379–756
Issue 1, 1–377

Volume 16, 10 issues

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author Index
To Appear
Other MSP Journals
This article is available for purchase or by subscription. See below.
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

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 a 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 C2 manifolds, up to a lower-dimensional subset.

PDF Access Denied

We have not been able to recognize your IP address as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

obstacle problem, fractional Laplacian, free boundary
Mathematical Subject Classification 2010
Primary: 35R35, 47G20
Received: 15 October 2019
Accepted: 5 February 2020
Published: 22 August 2021
Xavier Fernández-Real
Institute of Mathematics
Yash Jhaveri
Institute for Advanced Study
Princeton, NJ
United States