arXiv subjects
Publications and source records attributed to Akshara Vincent.
We slightly extend the convex integration via staircase laminate toolbox recently developed by Kleiner, M\"uller, Sz\'{e}kelyhidi, and Xie. As an example we revisit the proof by Astala-Faraco-Sz\'{e}kelyhidi on optimal Meyers' regularity theory via this framework.
We provide counterexamples to uniqueness of solutions as well as a priori Calder\'on-Zygmund estimates for solutions below $L^2$ using convex integration argument for equations of the type $$ \text{div} (A (\nabla u)) = 0 \quad \text{in } \mathbb{B}^2, $$ where $A: \mathbb{R}^{2} \to \mathbb{R}^2$ is smooth, uniformly elliptic and has essentially linear growth, but fails to be monotone and asymptotically Uhlenbeck.
We show that the mappings $u\in \dot{W}^{s,p}(\mathbb{R}^n,\mathcal{N})$ into manifolds $\mathcal{N}$ of a sufficiently simple topology that minimize the energy $$\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}} \;dx\;dy$$ are locally H\"older continuous in a bounded domain $\Omega$ outside a singular set $\Sigma $ with Hausdorff dimension strictly smaller than $n-sp$. We avoid the use of a monotonicity formula (which is unknown if $p \neq 2$) by using a blow-up argument instead.