Open Access
2016 A monotonicity formula for minimal sets with a sliding boundary condition
G. David
Publ. Mat. 60(2): 335-450 (2016). DOI: 10.5565/PUBLMAT_60216_04


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 $\varphi_t$ such that $\varphi_t(x) \in L$ 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.


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G. David. "A monotonicity formula for minimal sets with a sliding boundary condition." Publ. Mat. 60 (2) 335 - 450, 2016.


Received: 2 September 2014; Revised: 6 July 2015; Published: 2016
First available in Project Euclid: 11 July 2016

zbMATH: 1344.49073
MathSciNet: MR3521495
Digital Object Identifier: 10.5565/PUBLMAT_60216_04

Primary: 49K99 , 49Q20

Keywords: almost minimal sets , Minimal sets , monotonicity formula , Plateau problem , sliding boundary condition

Rights: Copyright © 2016 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.60 • No. 2 • 2016
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