• Calculus of Variations
• Free Boundary Problems
• Non-local Operators

This thesis aims to present a regularity result for the free boundary problem associated with the fractional Laplacian of order 3/2 . While we won’t explore the problem in its full generality, we will build upon the works of Aleksanyan and Anderssen by focusing on almost one-dimensional solutions. The main difficulty in the study of this problem is that the fractional Laplacian is a nonlocal operator. Therefore, its value in a point depends on the function’s behavior in the whole space. By leveraging the notable contributions of Caffarelli and Silvestre, as well as Yang and Cora-Musina, we can establish an equivalence between our problem and the bi-harmonic problem in the upper half-plane of R^{n+1}. Our ultimate objective is to show that the boundary of the positivity set, also known as the free boundary, exhibits C^{1,α} regularity. We begin by presenting the regularity results established by Danielli, Haj Ali, and Petrosyan.
These results indicate that if the biharmonic extension remains bounded in H1, then the solution itself remains bounded in H^{5/2} ∩ C^{1,1}. Building upon this outcome, we derive the conclusion that if uj represents a sequence of almost one-dimensional solutions with gradients converging to 0 in L^2, then uj converges to the function b · (xn)^{5/2}_+ where b is a positive constant. This result is to be expected, as this function stands as the unique solution of our problem when n = 1. Thanks to this outcome and employing the extension theorem introduced by Yang, we follow Aleksanyan’s approach to show that the free boundary can be locally represented as a graph of a C^{1,α} function. A natural question at this point is what happens if we consider different powers of the Laplace operator. One peculiarity of the case s = 3/2 is that Yang’s extension theorem yields to a bi-harmonic problem in R^{n+1}, and this is not true anymore if s ̸= 3/2 . Therefore, we are not able to find an explicit formula for the one-dimensional solution. To
overcome this problem, we could adopt a perturbative approach studying the case where s ∈ ( 3/2 − ε, 3/2 + ε).