Thin Obstacle-type Problems for the Bilaplacian

In this talk, we consider a class of obstacle-type problems involving a fourth-order elliptic operator. The possibly singular or degenerate operators we study arise naturally from extension procedures for higher-order fractional powers of the Laplacian and include two-phase boundary obstacle problems as a special case. After establishing well-posedness, we reformulate the problem as a coupled system of two second-order equations, featuring a possibly nonlinear and non-differentiable Neumann boundary coupling. We discuss regularity properties of solutions and investigate the structure of the associated free boundary.