Related papers: $C^{1+\alpha}$-Regularity for Two-Dimensional Almo…
It is, by now, classical that lattices in higher rank semisimple groups have various rigidity properties. In this work, we add another such rigidity property to the list: uniform stability with respect to the family of unitary operators on…
In the setting of Carnot groups, we are concerned with the rectifiability problem for subsets that have finite sub-Riemannian perimeter. We introduce a new notion of rectifiability that is, possibly, weaker than the one introduced by…
We establish a partial $C^{1,\alpha}$ regularity result for minimizers of the optimal $p$-compliance problem with length penalization in any spatial dimension $N\geq 2$, extending some of the results obtained in…
We prove that if $\alpha\in (0,1/2]$, then the packing dimension of a set $E\subset\mathbb{R}^2$ for which there exists a set of lines of dimension $1$ intersecting $E$ in dimension $\ge \alpha$ is at least $1/2+\alpha+c(\alpha)$ for some…
We extend the results of our recent preprint [arXiv: 1811.00515] into higher dimensions $n \geq 4$. For minimizing harmonic maps $u\in W^{1,2}(\Omega,\mathbb{S}^2)$ from $n$-dimensional domains into the two dimensional sphere we prove: (1)…
In this article we study minimal homeomorphisms(all orbits are dense) of the tori $T^{n},$ $n<5.$ The linear part of a homeomorphism $\phi $ of $T^{n}$ is the linear mapping $L$ induced by $\phi $ on the first homology group of $T^{n}$. It…
This article is devoted to developing a theory for effective kernel interpolation and approximation in a general setting. For a wide class of compact, connected $C^\infty$ Riemannian manifolds, including the important cases of spheres and…
In this article we establish $C^{3,\alpha}$-regularity of the reduced boundary of stationary points of a nonlocal isoperimetric problem in a domain $\Omega \subset \mathbb{R}^n$. In particular, stationary points satisfy the corresponding…
We first take into account variational problems with periodic boundary conditions, and briefly recall some sufficient conditions for a periodic solution of the Euler-Lagrange equation to be either a directional, a weak, or a strong local…
As is well-know, on an Arens regular Banach algebra all continuous functionals are weakly almost periodic. In this paper we show that $\ell^1$-bases which approximate upper and lower triangles of products of elements in the algebra produce…
In this paper we prove weak L^{1,p} (and thus C^{\alpha}) compactness for the class of uniformly mean-convex Riemannian n-manifolds with boundary satisfying bounds on curvature quantities, diameter, and (n-1)-volume of the boundary. We…
In this manuscript, we delve into the study of maps $u\in W^{1,2}(\Omega;\overline M)$ that minimize the Alt-Caffarelli energy functional $$ \int_\Omega (|Du|^2 + q^2 \chi_{u^{-1}(M)})\,dx, $$ under the condition that the image $u(\Omega)$…
We study the boundary regularity of almost minimal and quasiminimal sets that satisfy sliding boundary conditions. The competitors of a set $E$ are defined as $F = \varphi_1(E)$, where $\{ \varphi_t \}$ is a one parameter family of…
The Reifenberg theorem \cite{reif_orig} tells us that if a set $S\subseteq B_2\subseteq \mathbb R^n$ is uniformly close on all points and scales to a $k$-dimensional subspace, then $S$ is H\"older homeomorphic to a $k$-dimensional Euclidean…
The local behavior of the lowest order boundary element method on quasi-uniform meshes for Symm's integral equation and the stabilized hyper-singular integral equation on polygonal/polyhedral Lipschitz domains is analyzed. We prove local a…
For any $n\ge 2$, $\Omega\subset\rn$, and any given convex and coercive Hamiltonian function $H\in C^{0}(\rn)$, we find an optimal sufficient condition on $H$, that is, for any $c\in\mathbb R$, the level set $H^{-1}(c)$ does not contains…
Approximate lattices of Euclidean spaces, also known as Meyer sets, are aperiodic subsets with fascinating properties. In general, approximate lattices are defined as approximate subgroups of locally compact groups that are discrete and…
We consider three dimensional piecewise linear cones in $\mathbb{R}^4$ that are mass minimizing w.r.t. Lipschitz maps in the sense of \cite{almgren1976existence} as in \cite{Taylor76}. There are three that arise naturally by taking products…
We establish local $C^{1,\alpha}$-regularity for some $\alpha\in(0,1)$ and $C^{\alpha}$-regularity for any $\alpha\in(0,1)$ of local minimizers of the functional \[ v\ \mapsto\ \int_\Omega \phi(x,|Dv|)\,dx, \] where $\phi$ satisfies a…
We prove a multi-valued $C^{1,\alpha}$ regularity theorem for the varifolds in the class $\mathcal{S}_2$ (i.e., stable codimension one stationary integral $n$-varifolds admitting no triple junction classical singularities) which are…