A monotonicity formula for minimal sets with a sliding boundary condition

TL;DR

This paper establishes a monotonicity formula for minimal sets with sliding boundary conditions, enabling a better understanding of their local structure near the boundary in specific simple cases.

Contribution

It introduces a new monotonicity formula for minimal sets with sliding boundaries and applies it to describe their local behavior near the boundary.

Findings

Monotonicity formula for minimal sets with sliding boundary conditions.

Characterization of the set's structure when the monotone functional is constant.

Application of the formula to describe local behavior near the boundary.

Abstract

We prove a monotonicity formula for minimal or almost minimal sets for the Hausdorff measure $\cal{H}^d$, 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 $F(r) = r^{-d} \cal{H}^d(E \cap B(x,r)) + r^{-d} \cal{H}^d(S \cap B(x,r))$, 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. ----- On donne une formule de monotonie pour des ensembles minimaux ou presque minimaux pour la mesure de Hausdorff $\cal{H}^d$, avec une condition de bord o\`u les comp\'etiteurs de $E$ sont obtenus en d\'eformant $E$ par une famille \`a un param\`etre de fonctions $\varphi_t$ telles que $\varphi_t(x)\in L$ quand $x\in E$ se trouve sur la fronti\`ere $L$. Dans le cas simple o\`u $L$ est un sous-espace affine de dimension $d-1$, la fonctionelle monotone ou presque monotone est donn\'ee par $F(r) = r^{-d} \cal{H}^d(E \cap B(x,r)) + r^{-d} \cal{H}^d(S \cap B(x,r))$, o\`u $x$ est un point de $E$, pas forc\'ement dans $L$, et $S$ est l'ombre de $L$, \'eclair\'ee depuis $x$. On utilise ceci, la description des cas o\`u $F$ est constante, et un argument de limite, pour donner une description de $E$ pr\`es de $L$ dans deux cas simples.