C1+α-Regularity for Two-Dimensional Almost-Minimal Sets in ℝn
C1+α-Regularity for Two-Dimensional Almost-Minimal Sets in ℝn
复制标题
DOI：
10.1007/s12220-010-9138-z
复制
发表时间：
2010-05
影响因子：
1.1
通讯作者：
G. David
中科院分区：
文献类型：
--
作者：
G. David
我们给出了Jean Taylor的结果(ANN)的一个新证明和部分推广。数学课。(2)第103(3)，489-539,1976)指出ℝ3中维度为2的Almgren几乎极小集是局部等价于极小锥的C_1+α。证明相当简单，但使用了在ANN中证明的局部分离结果。FAC。SCI。Toulouse 18(1)，65-246,2009和Reifenberg的参数化定理的推广(David et al.在Geom。功能。肛门。18,1168-1235,2008)。其核心思想仍然是，如果X是单位球面上小Lipschitz图的圆弧上的锥体，但X不包含在圆盘中，我们可以利用调和函数的图形来变形X，使其面积大大减小。局部分离的结果被用来归结为Lipschitz图的弧上锥的并。证明的很大一部分推广到了ℝn中的2维极小集，但在这种情况下，我们最终的正则性结果E可能依赖于作为某一点的爆破极限而得到的极小锥的列表。
We give a new proof and a partial generalization of Jean Taylor’s result (Ann. Math. (2) 103(3), 489–539, 1976) that says that Almgren almost-minimal sets of dimension 2 in ℝ3are locallyC1+α-equivalent to minimal cones. The proof is rather elementary, but uses a local separation result proved in Ann. Fac. Sci. Toulouse 18(1), 65–246, 2009 and an extension of Reifenberg’s parameterization theorem (David et al. in Geom. Funct. Anal. 18, 1168–1235, 2008). The key idea is still that ifXis the cone over an arc of small Lipschitz graph in the unit sphere, butXis not contained in a disk, we can use the graph of a harmonic function to deformXand substantially diminish 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 ℝn, but in this setting our final regularity result onEmay depend on the list of minimal cones obtained as blow-up limits ofEat a point.