Related papers: H\"older Regularity of Two-Dimensional Almost-Mini…

A Besicovitch set is a set which contains a unit line segment in any direction. It is known that the Minkowski and Hausdorff dimensions of such a set must be greater than or equal to 5/2 in \R^3. In this paper we show that the Minkowski…

In this paper we study the local regularity of almost minimizers of the functional \begin{equation*} J(u)=\int_\Omega |\nabla u(x)|^2 +q^2_+(x)\chi_{\{u>0\}}(x) +q^2_-(x)\chi_{\{u<0\}}(x) \end{equation*} where $q_\pm \in L^\infty(\Omega)$.…

In 1960 Reifenberg proved the topological disc property. He showed that a subset of $R^n$ which is well approximated by $m$-dimensional affine spaces at each point and at each (small) scale is locally a bi-H\"older image of the unit ball in…

In this paper we consider nonlinear problems with an operator depending only on the deformation tensor. We consider the class of operators derived from a potential and with $(p,\delta)$ structure, for $1<p\leq 2$ and for all $\delta\geq0$.…

We prove quasiminimality of the regular mixed Tsirelson spaces T[(S_n,\theta_n)_n] with the sequence (\frac{\theta_n}{\theta^n})_n decreasing, where \theta=\lim_n \theta_n^{1/n}, and quasiminimality of all mixed Tsirelson spaces…

The well-known Reifenberg theorem states that if a subset of $\mathbb{R}^n$ can be well approximated by $k$-planes at every point and every scale, then it is biH\"older homeomorphic to a $k$-disk. This article concerns a subset $S$ of…

This paper discusses the regularity of multiple-valued Dirichlet minimizing maps into the sphere. It shows that even at branched point, as long as the normalized energy is small enough, we have the energy decay estimate. Combined with the…

In this article we prove that for a large class of 2-dimensional minimal cones (including almost all 2-dimensional minimal cones that we know), the almost orthogonal union of any two of them is still a minimal cone. Comparing to existing…

A regular partition $\mathcal{P}$ for a $3$-uniform hypergraph $H=(V,E)$ consists of a partition $V=V_1\cup \ldots \cup V_t$ and for each $ij\in {[t]\choose 2}$, a partition $K_2[V_i,V_j]=P_{ij}^1\cup \ldots \cup P_{ij}^{\ell}$, such that…

For a regular, compact, polynomially convex circled set K in C^2, we construct a sequence of pairs {P_n,Q_n} of homogeneous polynomials in two variables with deg P_n = deg Q_n = n such that the sets K_n: = {(z,w) \in C^2 : |P_n(z,w)| \leq…

We prove regularity results for the unique minimizer of the total variation functional, currently used in image processing analysis since the work by L. Rudin, S. Osher and E. Fatemi. In particular we show that if the source term $f$ is…

A Theorem of Kirichenko states that the torsion 3-form of the characteristic connection of a nearly K\"ahler manifold is parallel. On the other side, any almost hermitian manifold of type $\mathrm{G}_1$ admits a unique connection with…

We define a natural notion of higher order stability and show that subsets of $\mathbb{F}_p^n$ that are tame in this sense can be approximately described by a union of low-complexity quadratic varieties, up to linear error. This generalizes…

We prove a quantitative estimate on the number of certain singularities in almost minimizing clusters. In particular, we consider the singular points belonging to the lowest stratum of the Federer-Almgren stratification (namely, where each…

In this paper, we study the regularity of the "regular" part of the free boundary for almost minimizers in the parabolic Signorini problem with zero thin obstacle. This work is a continuation of our earlier research on the regularity of…

Let $(X,d)$ be a compact metric space, $f:X \mapsto X$ be a continuous map satisfying a property we call almost specification (which is slightly weaker than the $g$-almost product property of Pfister and Sullivan), and $\phi$ be a…

We study the properties of sets $\Sigma$ having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets $\Sigma \subset \mathbb{R}^2$ satisfying the inequality $\mbox{max}_{y \in M}…

We show that if $M^n$ is a properly immersed, two-sided, stable minimal hypersurface in $B^{n+1}_1(0)\setminus S$, where $S$ is closed with $\mathcal{H}^{n-2}(S)=0$, then $\text{dim}_{\mathcal{H}}\text{sing}(M)\leq n-7$, namely…

We study the regularity and the algebraic properties of certain lattice ideals. We establish a map I --> I\~ between the family of graded lattice ideals in an N-graded polynomial ring over a field K and the family of graded lattice ideals…

That is, given a compact set $B \subset \mathbb{R}^n$ (the boundary) and a subgroup $L$ of the \v{C}ech homology group $\check{H}_{d-1}(B;G)$ of dimension $d$ over some commutative group $G$, we find a compact set $E \supset B$ such that…