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

Abstract

Building upon the recent results in [15] we provide a thorough description of the free boundary for solutions to the fractional obstacle problem in R n+1 with obstacle function ϕ (suitably smooth and decaying fast at infinity) up to sets of null H n−1 measure. In particular, if ϕ is analytic, the problem reduces to the zero obstacle case dealt with in [15] and therefore we retrieve the same results:(i) local finiteness of the (n − 1)-dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure), (ii) H n−1 -rectifiability of the free boundary, (iii) classification of the frequencies and of the blow-ups up to a set of Hausdorff dimension at most (n − 2) in the free boundary. Instead, if ϕ ∈ C k+1 (R n ), k ≥ 2, similar results hold only for distinguished subsets of points in the free boundary where the order of contact of the solution with the obstacle function ϕ is less than k + 1.2010 Mathematics Subject Classification. Primary 35R35, 49Q20.

Direct Epiperimetric Inequalities for the Thin Obstacle Problem and Applications

“…Using this example one can easily construct global solutions in any dimension d 2 whose entire free boundary is a .d 2/-dimensional plane consisting only of points with frequency 2m 1 =2. Recently Focardi and Spadaro [9,10] proved the H d 2 -rectifiability of the set Other.u/ and that it consists of points of frequency 2m 1 =2 up to a set of zero H d 2 measure, but nothing has been known up to now regarding its regularity in dimension d > 2. We notice that in some special cases, the set Other.u/ might be empty.…”

“…A very important and not yet well understood question in the context of the thin-obstacle problem is the study of the admissible frequencies at free-boundary points. Indeed, nothing is known except for the gap between 3 =2 and 2 (see [2]) and the recent result of Focardi and Spadaro [9,10], where they establish that the collection of free-boundary points with frequency other than 3 =2, 2m, and 2m 1 =2 is a set of H d 2 measure zero. It is conjectured that these are the only admissible frequencies, but not even the gap between 2 and the subsequent admissible frequency was known.…”

Free boundary regularity in obstacle problems

“…Let k be the smallest dimension for which there exists a nontrivial solution to the problem above with λ * ∈ (2, 3). Then dim H (Σ a n−1 ) ≤ n − k. Because k ≥ 3 is the best lower bound currently known on k (see for instance [16]), we get dim H (Σ a n−1 ) ≤ n − 3.…”

“…Chapter 2.2.3, Theorem 7]): if ∆w = 0 inside B r (x), then(16) r|∇w(x)| + r 2 |D 2 w(x)| ≤ C(n) w L ∞ (Br(x)).Let x ∈ {v > 0} ∩ Ω, assume that dist(x, ∂{v > 0}) ≤ dist(x, ∂Ω), and set r := dist(x, ∂{v > 0}). Consider the function w(y) := u(y) − ϕ(x) − ∇ϕ(x) · (y − x)…”

On the Singular Set of Free Interface in an Optimal Partition Problem

“…The analysis in both [20] and [3] essentially concerns the preimage of the vertex of a conical target, while the specific features differ due to different properties of respective targets and blowup maps. Likewise, Focardi and Spadaro adapted the techniques of [20] to thin obstacle problems in [24,25], where they study the free boundary in its entirety. In contrast, we do not consider the preimage of 0 P N as a whole in this article.…”