A monotonicity formula for minimal sets with a sliding boundary condition. (English) Zbl 1344.49073

David, G.

Publ. Mat., Barc. 60, No. 2, 335-450 (2016).

Summary: We prove a monotonicity formula for minimal or almost minimal sets for the Hausdorff measure \(\mathcal{H}^d\), subject to a sliding boundary constraint where competitors for \(E\) are obtained by deforming \(E\) by a one-parameter family of functions \(\phi_t\) such that \(\phi_t(x) \in L\) when \(x\in E\) 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 \(F(r) = r^{-d} \mathcal{H}^d(E \cap B(x,r)) + r^{-d} \mathcal{H}^d(S \cap B(x,r))\), 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.

MSC:

49Q20	Variational problems in a geometric measure-theoretic setting
28A78	Hausdorff and packing measures

Keywords:

minimal sets; almost minimal sets; Hausdorff measure; monotonicity formula; sliding boundary condition; Plateau problem