A monotonicity formula for minimal sets with a sliding boundary condition

We prove a monotonicity formula for minimal or almost minimal sets for the Hausdorff measure Hd, subject to a sliding boundary constraint where competitors for E are obtained by deforming E by a one-parameter family of functions yt such that yt(x) ∈ L when x ∈ E lies on the boundary L. In the simple case when L is an affine subspace of dimension d-1, the monotone or almost monotone functional is given by F(r) = r-d Hd (E∩B(x, r)) + r-d Hd (S∩B(x,r)) where x is any point of E (not necessarily on L) and S is the shade of L with a light at x. We then use this, the description of the case when F is constant, and a limiting argument, to give a rough description of E near L in two simple cases