$C^{1+\alpha}$-REGULARITY FOR TWO-DIMENSIONAL ALMOST-MINIMAL SETS IN $\R^n$

Guy David (1)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay

We give a new proof and a partial generalization of Jean Taylor's result [Ta] that says that Almgren almost-minimal sets of dimension $2$ in $\R^3$ are locally $C^{1+\alpha}$-equivalent to minimal cones. The proof is rather elementary, but uses a local separation result proved in [D3] and an extension of Reifenberg's parameterization theorem [DDT]. The key idea is still that if $X$ is the cone over an arc of small Lipschitz graph in the unit sphere, but $X$ is not contained in a disk, we can use the graph of a harmonic function to deform $X$ and diminish substantially its area. The local separation result is used to reduce to unions of cones over arcs of Lipschitz graphs. A good part of the proof extends to minimal sets of dimension $2$ in $\R^n$, but in this setting our final regularity result on $E$ may depend on the list of minimal cones obtained as blow-up limits of $E$ at a point.

Mots clés: Hausdorff measure; Minimal sets; Almgren restricted or quasiminimal sets; Epiperimetric inequalities; Hausdorff measure.

Domaines: Analyse classique [math.CA]

Guy David. $C^{1+\alpha}$-REGULARITY FOR TWO-DIMENSIONAL ALMOST-MINIMAL SETS IN $\R^n$. 2008. ⟨hal-00287628⟩