no code implementations • 26 Jan 2021 • Katarzyna Mazowiecka, Jean Van Schaftingen
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 Borel-measurable maps $u\colon \partial \mathcal M \to \mathcal{N}$ that are the trace of a map $U\in \dot{W}^{1, p}(\mathcal M, \mathcal{N})$ are characterized as the maps for which there exists an extension energy density $w \colon \partial \mathcal{M} \to [0,\infty]$ that controls the Sobolev energy of extensions from $\lfloor p - 1 \rfloor$-dimensional subsets of $\partial \mathcal{M}$ to $\lfloor p\rfloor$-dimensional subsets of $\mathcal{M}$.
Analysis of PDEs Functional Analysis
no code implementations • 18 Jan 2021 • Francesca Da Lio, Katarzyna Mazowiecka, Armin Schikorra, LiFeng Wang
We prove that for antisymmetric vectorfield $\Omega$ with small $L^2$-norm there exists a gauge $A \in L^\infty \cap \dot{W}^{1/2, 2}(\mathbb{R}^1, GL(N))$ such that ${\rm div}_{\frac12} (A\Omega - d_{\frac{1}{2}} A) = 0$.
Analysis of PDEs
