The local structure of the free boundary in the fractional obstacle problem

2019/03/14 by Focardi, Matteo, Spadaro, Emanuele
#35R35 #49Q20 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1903.05909

Abstract

Building upon the recent results in \citeFoSp17 we provide a thorough description of the free boundary for the fractional obstacle problem in ℝⁿ⁺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 \citeFoSp17 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(ℝⁿ), k≥ 2, similar results hold only for a distinguished subset of points in the free boundary where the order of contact of the solution and the obstacle is less than k+1.

Related

• Generic regularity of free boundaries in the obstacle problem for the fractional Laplacian
• Global regularity for the free boundary in the obstacle problem for the fractional Laplacian
• Higher Regularity for the fractional thin obstacle problem
• The structure of the regular part of the free boundary close to singularities in the obstacle problem
• Structure and regularity of the singular set in the obstacle problem for the fractional Laplacian