A One-phase Problem for the Fractional Laplacian: Regularity of Flat Free Boundaries

Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian

We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem. We write an equivalent characterization as a thin obstacle problem. In this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary.

The N-Membranes Problem

Existence solutions for new p-Laplacian fractional boundary value problem with impulsive effects

Fractional differential equations have been of great interest recently. This is because of both the intensive development of the theory of fractional calculus itself and the applications of such constructions in various scientific fields such as physics, mechanics, chemistry, engineering, etc. Differential equations with impulsive effects arising from the real world describe the dyn...

Global Regularity for the Free Boundary in the Obstacle Problem for the Fractional Laplacian

We study the regularity of the free boundary in the obstacle problem for the fractional Laplacian under the assumption that the obstacle φ satisfies ∆φ ≤ 0 near the contact region. Our main result establishes that the free boundary consists of a set of regular points, which is known to be a (n− 1)-dimensional C manifold by the results in [7], and a set of singular points, which we prove to be c...

ON THE REGULARITY OF THE FREE BOUNDARY IN THE p-LAPLACIAN OBSTACLE PROBLEM

We study the regularity of the free boundary in the obstacle for the p-Laplacian, min { −∆pu, u − φ } = 0 in Ω ⊂ R. Here, ∆pu = div ( |∇u|p−2∇u ) , and p ∈ (1, 2) ∪ (2,∞). Near those free boundary points where ∇φ 6= 0, the operator ∆p is uniformly elliptic and smooth, and hence the free boundary is well understood. However, when ∇φ = 0 then ∆p is singular or degenerate, and nothing was known ab...