Related papers: Trace theory for Sobolev mappings into a manifold

The compact Riemannian manifolds $\mathcal{M}$ and $\mathcal{N}$ for which the trace operator from the first-order Sobolev space of mappings $\smash{\dot{W}}^{1, p} (\mathcal{M}, \mathcal{N})$ to the fractional Sobolev-Slobodecki\u{\i}…

We give direct proofs and constructions of the trace and extension theorems for Sobolev mappings in $W^{1, 1} (M, N)$, where $M$ is Riemannian manifold with compact boundary $\partial M$ and $N$ is a complete Riemannian manifold. The…

The main result of the present paper, combined with earlier results of Hardt and Lin settles the extension problem for $W^{1,p}(\mathcal M, \mathcal N)$, where $\mathcal M$ and $\mathcal N$ are compact riemannian manfolds, $\mathcal M$…

Motivated by manifold-constrained homogenization problems, we construct suitable extensions for Sobolev functions defined on a perforated domain and taking values in a compact, connected $C^2$-manifold without boundary. The proof combines a…

We give a quantitative characterization of traces on the boundary of Sobolev maps in $\dot{W}^{1,p}(\mathcal M, \mathcal N)$, where $\mathcal{M}$ and $\mathcal{N}$ are compact Riemannian manifolds, $\partial \mathcal{M} \neq \emptyset$: the…

We prove that a mapping $u \colon \mathcal{M}'\to \mathcal{N}$, where $\mathcal{M}'$ and $ \mathcal{N}$ are compact Riemannian manifolds, is the trace of a Sobolev mapping $U \colon \mathcal{M}' \times [0, 1) \to \mathcal{N}$ if and only if…

We prove that if $M\subset \mathbb{R}^n$ is a bounded subanalytic submanifold of $\mathbb{R}^n$ such that $B(x_0,\epsilon)\cap M$ is connected for every $x_0\in\overline{M}$ and $\epsilon>0$ small, then, for $p\in [1,\infty)$ sufficiently…

Let $S \subset \mathbb{R}^{n}$ be an arbitrary nonempty compact set such that the $d$-Hausdorff content $\mathcal{H}^{d}_{\infty}(S) > 0$ for some $d \in (0,n]$. For each $p \in (\max\{1,n-d\},n]$, an almost sharp intrinsic description of…

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~$x \in S$…

Trace classes of Sobolev-type functions in metric spaces are subject of this paper. In particular, functions on domains whose boundary has an upper codimension-$\theta$ bound are considered. Based on a Poincar\'e inequality, existence of a…

Given $n \in \mathbb{N}_*$, a compact Riemannian manifold $M$ and a Sobolev map $u \in W^{n/(n + 1), n + 1} (\mathbb{S}^n; M)$, we construct a map $U$ in the Sobolev-Marcinkiewicz (or Lorentz-Sobolev) space $W^{1, (n + 1, \infty)}…

We focus on the Sobolev spaces of bounded subanalytic submanifolds of $\mathbb{R}^n$. We prove that if $M$ is such a manifold then the space $\mathscr{C}_0^\infty(M)$ is dense in $W^{1,p}(M,\partial M)$ (the kernel of the trace operator)…

In this paper we fix $1\le p<\infty$ and consider $(\Om,d,\mu)$ be an unbounded, locally compact, non-complete metric measure space equipped with a doubling measure $\mu$ supporting a $p$-Poincar\'e inequality such that $\Om$ is a uniform…

Let $(\operatorname{X},\operatorname{d},\mu)$ be a metric measure space with uniformly locally doubling measure $\mu$. Given $p \in (1,\infty)$, assume that $(\operatorname{X},\operatorname{d},\mu)$ supports a weak local $(1,p)$-Poincar\'e…

We give a sharp characterization of how additional integrability in the interior improves the integrability of boundary traces of $\mathrm{W}^{1,p}$-Sobolev functions. The optimality of our results relies on a novel nonlinear extension or…

For any integer $ p \geq 2 $, we construct a compact Riemannian manifold $ \mathcal{N} $ such that if $ \dim \mathcal{M} > p $, there is a map in the Sobolev space of mappings $ W^{1,p} (\mathcal{M}, \mathcal{N})$ which is not a weak limit…

A complete description of traces on $\mathbb{R}^{n}$ of functions from the weighted Sobolev space $W^{l}_{1}(\mathbb{R}^{n+1},\gamma)$, $l \in \mathbb{N}$, with weight $\gamma \in A^{\rm loc}_{1}(\mathbb{R}^{n+1})$ is obtained. In the case…

Given $m \in \mathbb{N} \setminus \{0\}$ and a compact Riemannian manifold $\mathcal{N}$, we construct for every map $u$ in the critical Sobolev space $W^{m/(m + 1), m + 1} (\mathbb{S}^m, \mathcal{N})$, a map $U : \mathbb{B}^{m + 1} \to…

The focus of this paper is on Ahlfors $Q$-regular compact sets $E\subset\mathbb{R}^n$ such that, for each $Q-2<\alpha\le 0$, the weighted measure $\mu_{\alpha}$ given by integrating the density $\omega(x)=\text{dist}(x, E)^\alpha$ yields a…

Given $p \in (1,\infty)$, let $(\operatorname{X},\operatorname{d},\mu)$ be a metric measure space with uniformly locally doubling measure $\mu$ supporting a weak local $(1,p)$-Poincar\'e inequality. For each $\theta \in [0,p)$, we…