We will describe the lower-dimensional obstacle problem for a uniformly elliptic, divergence form operator with Lipschitz continuous coefficients. We will give an overview of what is known about this problem, new developments and the role of a new monotonicity formula for an appropriate generalization of Almgren's frequency functional in the optimal regularity of the solution.

The main point of the lecture should concern regularity properties of minimal or almost minimal sets that satisfy a sliding boundary condition of Plateau type, and in particular near the boundary. We should in particular discuss some monotonicity property of density, and simple applications in low dimensions.