$C^{1+α}$-Regularity for Two-Dimensional Almost-Minimal Sets in $\R^n$
| dc.creator | David, Guy | |
| dc.date | 2008-06-12 | |
| dc.date.accessioned | 2026-07-07T12:19:30Z | |
| dc.date.available | 2026-07-07T12:19:30Z | |
| dc.description | 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 $C^{1+α}$-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. | |
| dc.description | 115 pages, 4 figures | |
| dc.identifier | https://arxiv.org/abs/0806.2080 | |
| dc.identifier | http://arxiv.org/abs/0806.2080 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/212776 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | 49K99 ; 49Q20 | |
| dc.title | $C^{1+α}$-Regularity for Two-Dimensional Almost-Minimal Sets in $\R^n$ | |
| dc.type | text |