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

Abstract

In this work, 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 non-local 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 two 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 aharmonic polynomials when a ≥ 0. To do so, we establish regularity results for this codimension two problem, what 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.

On the Measure and the Structure of the Free Boundary of the Lower Dimensional Obstacle Problem

“…The latter result has been very recently extended to the full range a ∈ (−1, 1) and to ϕ ∈ C k+1 (R n ), k ≥ 2, in [18]. Furthermore, fine properties of the singular set have been studied very recently by Fernández-Real and Jhaveri [13]. It is also worth mentioning the paper by Barrios, Figalli and Ros-Oton [3], in which the authors study the fractional obstacle problem (1.1) with non zero obstacle ϕ having compact support and satisfying suitable concavity assumptions.…”

The structure of the singular set in the thin obstacle problem for degenerate parabolic equations

“…More recently, their results have been extended to the whole range a ∈ (−1, 1) in [23]. We also mention the recent interesting paper [13], where for the time-independent Signorini problem (a = 0) a finer stratification of the singular set is obtained using a variant of Weiss' epiperimetric inequality, and the work [17] for a further refined analysis of the structure of the singular set under certain geometric assumption on the obstacle. A parabolic version of such epiperimetric inequality (again, when a = 0) has been very recently established in [29], where it has also been shown that such an inequality, combined with the results in [15], provides a finer structure theorem of the singular set in the parabolic thin obstacle problem.…”