  In this paper we are concerned with a two phase boundary obstacle-type
problem for the bi-Laplace operator in the upper unit ball. The problem arises
in connection with unilateral phenomena for flat elastic plates. It can also be
seen as an extension problem to an obstacle problem for the fractional
Laplacian $(-\Delta )^{3/2}$, as first observed in \cite{Y}. We establish the
well-posedness and the optimal regularity of the solution, and we study the
structure of the free boundary. Our proofs are based on monotonicity formulas
of Almgren- and Monneau-type.
