arXiv subjects

Pawel Goldstein

Publications and source records attributed to Pawel Goldstein.

3 recordsLinked to original sources

arXiv2018

Uhlenbeck's decomposition in Sobolev and Morrey-Sobolev spaces

We present a self-contained proof of Uhlenbeck's decomposition theorem for $Ω\in L^p(\mathbb{B}^n,so(m)\otimesΛ^1\mathbb{R}^n)$ for $p\in (1,n)$ with Sobolev type estimates in the case $p \in[n/2,n)$ and Morrey-Sobolev type estimates in the case $p\in (1,n/2)$. We also prove an analogous theorem in the case when $Ω\in L^p( \mathbb{B}^n, TCO_{+}(m) \otimes Λ^1\mathbb{R}^n)$, which corresponds to Uhlenbeck's theorem with conformal gauge group.

math.AP↗

arXiv2017

Characterizations of generalized John domains in $\mathbb{R}^n$ via metric duality

In this paper, we extend the characterization of John disks obtained by N\"akki and V\"ais\"al\"a [Exp. Math. 1991] to generalized John domains in higher dimensions under mild assumptions. The main ingredient in this characterization is to use the higher dimensional analogues of the local linear connectivity (LLC) and homological bounded turning properties introduced by V\"ais\"al\"a in his study of metric duality theory [Math. Scan. 1997]. Somewhat surprisingly, we constructed a uniform domain in $\R^3$, which is topologically simple, such that the complementary domain fails to be homotopically $1$-bounded turning. In particular, this shows that a similar characterization of generalized John domains in terms of higher dimensional homotopic bounded turning does not hold in dimension three.

math.GN↗

arXiv2011

Sobolev mappings, degree, homotopy classes and rational homology spheres

In the paper we investigate the degree and the homotopy theory of Orlicz-Sobolev mappings $W^{1,P}(M,N)$ between manifolds, where the Young function $P$ satisfies a divergence condition and forms a slightly larger space than $W^{1,n}$, $n=\dim M$. In particular, we prove that if $M$ and $N$ are compact oriented manifolds without boundary and $\dim M=\dim N=n$, then the degree is well defined in $W^{1,P}(M,N)$ if and only if the universal cover of $N$ is not a rational homology sphere, and in the case $n=4$, if and only if $N$ is not homeomorphic to $S^4$.

math.FA↗