Related papers: The Relative Capacity

We study the approximative trace for individual elements in the Sobolev space $W^{1,p}(\Omega)$ for $1\le p\le\infty$. This notion of a trace was introduced for $p=2$ in [AtE11] in the setting of general open sets…

We consider integral inequalities in the sense of Choquet with respect to the Hausdorff content $\mathcal{H}_\infty^{\delta}$. In particular, if $\Omega$ is a bounded John domain in $\mathbb{R}^n$, $n\geq 2$, and $0 <\delta \le n$, we prove…

We are concerned here with Sobolev-type spaces of vector-valued functions. For an open subset $\Omega\subset\mathbb{R}^N$ and a Banach space $V$, we compare the classical Sobolev space $W^{1,p}(\Omega, V)$ with the so-called…

In this paper we study approximations of functions of Sobolev spaces $W^2_{p,\loc}(\Omega)$, $\Omega\subset\mathbb R^n$, by Lipschitz continuous functions. We prove that if $f\in W^2_{p,\loc}(\Omega)$, $1\leq p<\infty$, then there exists a…

We investigate the connection between measure, capacity and algorithmic randomness for the space of closed sets. For any computable measure m, a computable capacity T may be defined by letting T(Q) be the measure of the family of closed…

Let $\Omega\subset\mathbb{R}^n$ be an open set and let $f\in W^{1,p}(\Omega,\mathbb{R}^n)$ be a weak (sequential) limit of Sobolev homeomorphisms. Then $f$ is injective almost everywhere for $p>n-1$ both in the image and in the domain. For…

This paper studies the inclusions between different Sobolev-Lorentz spaces $W^{1,(p,q)}(\Omega)$ defined on open sets $\Omega \subset {\mathbf{R}^n},$ where $n \ge 1$ is an integer, $1<p<\infty$ and $1 \le q \le \infty.$ We prove that if $1…

We consider different notions of capacity related to the parabolic $p$-Laplace equation. Our focus is on a variational notion, which is consistent in the full range $1<p<\infty$. For such a notion we show some basic properties as well as…

Let $\Omega$ be a bounded, smooth domain of $\mathbb{R}^{N},$ $N\geq2.$ For $1<p<N$ and $0<q(p)<p^{\ast}:=\frac{Np}{N-p}$ let \[ \lambda_{p,q(p)}:=\inf\left\{ \int_{\Omega}\left\vert \nabla u\right\vert ^{p}\mathrm{d}x:u\in…

For each $p>n$ we use local oscillations and doubling measures to give intrinsic characterizations of the restriction of the Sobolev space $W_p^1(R^n)$ to an arbitrary closed subset of $R^n$.

In the Euclidean setting the Sobolev spaces $W^{\alpha,p}\cap L^\infty$ are algebras for the pointwise product when $\alpha>0$ and $p\in(1,\infty)$. This property has recently been extended to a variety of geometric settings. We produce a…

We describe a class of Sobolev $W^k_p$-extension domains $\Omega\subset R^n$ determined by a certain inner subhyperbolic metric in $\Omega$. This enables us to characterize finitely connected Sobolev $W^1_p$-extension domains in $R^2$ for…

In this paper we give an elementary proof that sets of zero $p,1$-Sobolev-Lorentz capacity are $\mathcal{H}^{n-p}$-null sets independently of non-linear potential theory. We further show that there exists a set of Sobolev-Lorentz-$(p,1)$…

For each $p>1$ and each positive integer $m$ we use divided differences to give intrinsic characterizations of the restriction of the Sobolev space $W^m_p(R)$ to an arbitrary closed subset of the real line.

Let $\Omega$ be an open set in a metric measure space $X$. Our main result gives an equivalence between the validity of a weighted Hardy-Sobolev inequality in $\Omega$ and quasiadditivity of a weighted capacity with respect to Whitney…

Let $\Omega \subset \mathbb{R}^n$ be an open set and $f_k \in W^{s,p}(\Omega;\mathbb{R}^n)$ be a sequence of homeomorphisms weakly converging to $f \in W^{s,p}(\Omega;\mathbb{R}^n)$. It is known that if $s=1$ and $p > n-1$ then $f$ is…

Let $\Omega$ be a bounded domain of $\mathbf{R}^{N},$ $N\geq2.$ Let, for $p>N,$ \[ \Lambda_{p}(\Omega):=\inf\left\{ \left\Vert \nabla u\right\Vert _{p}^{p}:u\in W_{0}^{1,p}(\Omega)\quad and\quad\left\Vert u\right\Vert _{\infty}=1\right\} .…

Let $S \subset \mathbb{R}^{n}$ be a~closed set such that for some $d \in [0,n]$ and $\varepsilon > 0$ the~$d$-Hausdorff content $\mathcal{H}^{d}_{\infty}(S \cap Q(x,r)) \geq \varepsilon r^{d}$ for all cubes~$Q(x,r)$ centered in~$x \in S$…

The variational capacity cap_p in Euclidean spaces is known to enjoy the density dichotomy at large scales, namely that for every subset E of R^n, inf_{x in R^n} (cap_p(E \cap B(x,r),B(x,2r)) / cap_p(B(x,r),B(x,2r))) is either zero or tends…

We introduce a notion of maximal potentials and we prove that they form bounded operators from $L^p$ to the homogeneous Sobolev space $\dot{W}^{1,p}$ for all $n/(n-1)<p<n$. We apply this result to the problem of boundedness of the spherical…