11 problems

Let $\Omega\subset\mathbb R^n$ be a domain, let $F\in W^{1,n}_{\mathrm{loc}}(\Omega;\mathbb R^n)$ be a nonconstant sense-preserving mapping with dilatation…

Let $n\geq 2$, let $\Omega\subset\mathbb R^n$ be a domain, and let $F\colon\Omega\to\mathbb R^n$ be an orientation-preserving mapping in…

Higher-dimensional openness and discreteness conjecture. In higher dimensions, every non-constant mapping $f\in W^{1,n}_{\mathrm{loc}}(\Omega,\mathbb{R}^n)$ of finite distortion wi…

Let ${\Bbb D}$ be the unit disk and ${\Bbb S}=\partial{\Bbb D}$ its boundary. Suppose ${\cal A}:[1,\infty)\to[1,\infty)$ is convex and increasing, and for a mapping…

Let $\Omega$ and $\Omega'$ be domains in $\mathbb{R}^n$ with $n\geq 3$, let $f_0\colon\Omega\to\Omega'$ be a given homeomorphism, and let…

Ratio-integrability continuity conjecture. Then $f$ has a continuous representative. This is the explicit formulation of the proposed stronger continuity statement for generalized…

Let $\Omega\subset\mathbb{R}^n$ be a domain, let $1\leq p\leq\infty$, and let $p^*$ denote the Hölder conjugate of $p$. Consider $f\in W^{1,n}_{\mathrm{loc}}(\Omega,\mathbb{R}^n)$…

Let $\Omega\subset\mathbb{R}^n$ be a domain, let $1\leq p\leq\infty$, and let $p^*$ denote the Hölder conjugate of $p$. Consider a Sobolev mapping…

Let ${\Bbb D}$ be the unit disk, and let $f_0:\overline{{\Bbb D}}\to\overline{{\Bbb D}}$ be a homeomorphism of finite distortion with … Here ${\cal A}:[1,\infty)\to[1,\infty)$ is c…

Let $\Omega\subset \mathbb R^n$ be a bounded open connected domain with $n\geq 3$. Let $u\in W^{1,n}(\Omega)$ and $v\in W^{1,1}(\Omega)$ satisfy … almost everywhere, and suppose th…

Let $\Omega\subset\mathbb{R}^n$ be a domain, let $f\in W^{1,n}_{\operatorname{loc}}(\Omega,\mathbb{R}^n)$, and let $K_I(\cdot,f)$ denote the inner distortion of $f$. Assume that…