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

Abstract

We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight |y| a for a ∈ (−1, 1). Such problem arises as the local extension of the obstacle problem for the fractional heat operator (∂t − ∆x) s for s ∈ (0, 1). Our main result establishes the complete structure and regularity of the singular set of the free boundary. To achieve it, we prove Almgren-Poon, Weiss, and Monneau type monotonicity formulas which generalize those for the case of the heat equation (a = 0).

The regular free boundary in the thin obstacle problem for degenerate parabolic equations

“…up to the thin manifold {y = 0}, for some α 0 > 0. Using such Hölder regularity of ∇ x U, y a U y , and the boundedness of U t , we can at this point argue as in the proof of Lemma 5.1 in [7], and conclude that the following W 2,2 type estimate holds for ρ < 1, (3.15)…”

“…We recall the definition (3.6) of κ 0 . The following gap result states, in particular, that κ 0 is the lowest possible frequency, see [7,Lemma 7.2]. Lemma 4.3.…”

“…Subsequently, we prove localized regularity estimates for the derivatives of U independent of the boundary conditions. Such local estimates are critical in the blowup analysis in Section 6 in [7], where the structure of the singular set is studied. The reader should be aware that we will often pass from a problem in Q + 1 to one in Q 1 , while keeping the same notation for the data of the problem.…”

“…For the case a ∈ (−1, 1), a version of the problem (1.2)-(1.3) has been recently studied in [2], where the authors used global assumptions on initial data ϕ 0 to infer quasi-convexity properties of the solutions, leading to their optimal regularity, as well as to the regularity of the free boundary near certain types of points. The approach that we take in the present paper is purely local, which is of crucial importance in the further analysis of the problem as it allows to consider the blowups at free boundary points, leading to their fine classification, see our forthcoming paper [7].…”

“…While in this paper we restrict ourselves to the study of regular free boundary points, in the forthcoming paper [7] we take a more systematic parabolic approach to the classification of free boundary points based on an Almgren-Poon type monotonicity formula. In that paper we also prove a structural theorem on the so-called singular set of the free boundary with the help of Weiss and Monneau type monotonicity formulas.…”