Related papers: $\Omega$-symmetric measures and related singular i…

In this paper it is shown that if $\mu$ is an n-dimensional Ahlfors-David regular measure in $R^d$ which satisfies the so-called weak constant density condition, then $\mu$ is uniformly rectifiable. This had already been proved by David and…

In this paper it is shown that if $\mu$ is a finite Radon measure in $\mathbb R^d$ which is $n$-rectifiable and $1\leq p\leq 2$, then $$\int_0^\infty \beta_{\mu,p}^n(x,r)^2\,\frac{dr}r<\infty \quad {for $\mu$-a.e. $x\in\mathbb R^d$,}$$…

We assume that $\Omega \subset \mathbb{R}^{d+1}$, $d \geq 2$, is a uniform domain with lower $d$-Ahlfors-David regular and $d$-rectifiable boundary. We show that if $\mathcal{H}^d|_{\partial \Omega}$ is locally finite, then the Hausdorff…

Let $\Omega \subset \mathbb{R}^{n+1}$, $n\geq 2$, be 1-sided NTA domain (aka uniform domain), i.e. a domain which satisfies interior Corkscrew and Harnack Chain conditions, and assume that $\partial\Omega$ is $n$-dimensional Ahlfors-David…

For a Radon measure $\mu$ on $\mathbb{R}^d$, define $C^n_\mu(x, t)= \ (\frac{1}{t^n} \ |\int_{B(x,t)} \frac{x-y}{t} \, d\mu(y)\ | \ )$. This coefficient quantifies how symmetric the measure $\mu$ is by comparing the center of mass at a…

In the following paper, we prove a dimension bound on the singular set of a Radon measure assuming its doubling ratio converges uniformly on compact sets. More precisely, we prove that if a Radon measure is $n$-Uniformly Asymptotically…

This paper is devoted to the proof of two related results. The first one asserts that if $\mu$ is a Radon measure in $\mathbb R^d$ satisfying $$\limsup_{r\to 0} \frac{\mu(B(x,r))}{r}>0\quad \text{ and }\quad…

A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures $\mu$ in $n$-dimensional Euclidean space for all $n\geq 2$ in terms of…

We provide a sufficient geometric condition for $\mathbb{R}^n$ to be countably $(\mu,m)$ rectifiable of class $\mathscr{C}^{1,\alpha}$ (using the terminology of Federer), where $\mu$ is a Radon measure having positive lower density and…

In the following paper, one studies, given a bounded, connected open set $\Omega$ $\subseteq$ R n , $\kappa$ > 0, a positive Radon measure $\mu$ 0 in $\Omega$ and a (signed) Radon measure $\mu$ on $\Omega$ satisfying $\mu$($\Omega$) = 0 and…

We study the regularity of Radon measures $\mu$ which satisfy that there exists a function $h_\mu$ in $H^1(\Omega)$, stationary harmonic such that $\Delta h_\mu =\mu$ in $\Omega$ (here $\Omega$ is an open set of $\mathbb{R}^2$). Such…

Given positive measures $\nu,\mu$ on an arbitrary measurable space $(\Omega, \mathcal F)$, we construct a sequence of finite partitions $(\pi_n)_n$ of $(\Omega, \mathcal F)$ s.t. $$ \sum_{A\in \pi_n: \mu(A)>0} 1_{A} \frac{\nu(A)}{\mu(A)}…

We study absolute continuity of harmonic measure with respect to surface measure on domains $\Omega$ that have large complements. We show that if $\Gamma\subset \mathbb{R}^{d+1}$ is $d$-Ahlfors regular and splits $ \mathbb{R}^{d+1}$ into…

In this paper, we prove the existence of multiple nontrivial solutions of the following equation. \begin{align*} \begin{split} -\Delta_{p}u & = \frac{\lambda}{u^{\gamma}}+g(u)+\mu~\mbox{in}\,\,\Omega, u & = 0\,\, \mbox{on}\,\,…

Let $\mu,\nu$ be Radon measures on $\mathbb{R}$, with $\mu$ non-atomic and $\nu$ doubling, and write $\mu = \mu_{a} + \mu_{s}$ for the Lebesgue decomposition of $\mu$ relative to $\nu$. For an interval $I \subset \mathbb{R}$, define…

We give a partial uniqueness result concerning comparable renormalized solutions of the nonlinear elliptic problem $-\diw(\aop(x,Du))=\mu$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\mu$ is a Radon measure with bounded variation on…

A Radon measure $\mu$ is $n$-rectifiable if it is absolutely continuous with respect to $\mathcal{H}^n$ and $\mu$-almost all of $\text{supp}\,\mu$ can be covered by Lipschitz images of $\mathbb{R}^n$. In this paper we give two sufficient…

Let $\Omega\subset\mathbb{R}^n$, $n\ge 2$, be a bounded connected $C^2$ domain. For any unit vector $\nu\in\mathbb{R}^n$, let $T_{\lambda}^{\nu}=\{x\in\mathbb{R}^n:x\cdot\nu=\lambda\}$,…

We characterize Radon measures $\mu$ in $\mathbb{R}^{n}$ that are $d$-rectifiable in the sense that their supports are covered up to $\mu$-measure zero by countably many $d$-dimensional Lipschitz graphs and $\mu \ll \mathcal{H}^{d}$. The…

Let $\Omega\subset\mathbb R^{n+1}$, $n\geq1$, be a corkscrew domain with Ahlfors-David regular boundary. In this paper we prove that $\partial\Omega$ is uniformly $n$-rectifiable if every bounded harmonic function on $\Omega$ is…