Related papers: $C^{1+\alpha}$-Regularity for Two-Dimensional Almo…

We give a different and probably more elementary proof of a good part of Jean Taylor's regularity theorem for Almgren almost-minimal sets of dimension 2 in $\R^3$. We use this opportunity to settle some details about almost-minimal sets,…

In this article we prove that the union of two almost orthogonal planes in R4 is Almgren-minimal. This gives an example of a one parameter family of minimal cones, which is a phenomenon that does not exist in R3. This work is motivated by…

We adapt the method of Simon [JDG '93] to prove a $C^{1,\alpha}$-regularity theorem for minimal varifolds which resemble a cone $\bf{C}_0^2$ over an equiangular geodesic net. For varifold classes admitting a "no-hole" condition on the…

We discuss the global regularity for 2 dimensional minimal sets that are near a $\T$ set, that is, whether every global minimal set in $\R^n$ that looks like a $\T$ set at infinity is a $\T$ set or not. The main point is to use the…

In this article we prove the topological minimality of unions of several almost orthogonal planes of arbitrary dimensions. A particular case was proved in arXiv:1103.1468, where we proved the Almgren minimality (which is a weaker property…

In [15], Jean Taylor has proved a regularity theorem away from boundary for Almgren almost minimal sets of dimension two in $\mathbb{R}^{3}$. It is quite important for understanding the soap films and the solutions of Plateau's problem away…

We prove an Allard-type regularity theorem for free-boundary minimal surfaces in Lipschitz domains locally modelled on convex polyhedra. We show that if such a minimal surface is sufficiently close to an appropriate free-boundary plane,…

We show that if (u;K) is a minimizer of the Mumford-Shah functional in an open set of R^3, and if x, K and r > 0 are such that K is close enough to a minimal cone of type P (a plane), Y (three half planes meeting with 120 degrees angles) or…

We show that directed minimal cones in (n+1)-dimensional Euclidean space which have at most one singularity are - besides the trivial cases: empty set, whole space - half spaces. Using blow-up techniques, this result can be used to get…

We study the local regularity of sliding almost minimal sets of dimension 2 in $R^n$ , bounded by a smooth curve $L$. These are a good way to model soap films bounded by a curve, and their definition is similar to Almgren's. We aim for a…

We discuss the global regularity of 2 dimensional minimal sets that are near a union of two planes, and prove that every global minimal set in R^4 that looks like a union of two almost orthogonal planes at infinity is a cone. The main point…

We introduce a notion of non-local almost minimal boundaries similar to that introduced by Almgren in geometric measure theory. Extending methods developed recently for non-local minimal surfaces we prove that flat non-local almost minimal…

In this article we treat two closely related problems: 1) the upper semi continuity property for Almgren minimal sets in regions with regular boundary, which guanrantees that the uniqueness property is well defined; and 2) the Almgren…

We prove some epsilon regularity results for n-dimensional minimal two-valued Lipschitz graphs. The main theorems imply uniqueness of tangent cones and regularity of the singular set in a neighbourhood of any point at which at least one…

In this paper, we prove that the product of a paired calibrated set and a set of codimension 1 calibrated by a coflat calibration with small singularity set is Almgren minimal. This is motivated by the attempt to classify all possible…

We present some old and recent regularity results concerning minimal and almost minimal sets in domains of the Euclidean space. We concentrate on a sliding variant of Almgren's notion of minimality, which is well suited in the context of…

In this article we prove the measure stability for all 2-dimensional Almgren minimal cones in $\mathbb{R}^n$, and the Almgren (resp. topological) sliding stability for the 2-dimensional Almgren (resp. topological) minimal cones in…

We present regularity results for the crack set of a minimizer for the Griffith fracture energy, arising in the variational modeling of brittle materials. In the planar setting, we prove an epsilon-regularity theorem showing that the crack…

In the Heisenberg group $\mathbb{H}^1$, equipped with a left-invariant and not necessarily symmetric norm in the horizontal distribution, we provide examples of entire area-minimizing horizontal graphs which are locally Lipschitz in…

We provide several equivalent characterizations of locally flat, $d$-Ahlfors regular, uniformly rectifiable sets $E$ in $\mathbb{R}^n$ with density close to $1$ for any dimension $d \in \mathbb{N}$ with $1 \le d \le n-1$. In particular, we…