13 problems

Cox et al.'s conjecture. There exist constants $C_1,C_2>0$, possibly depending only on $n$, such that

Optimal growth rate conjecture. The optimal growth rate estimate given in Theorem holds if and only if $\gamma<2$ (thus $\theta<\frac{3\pi}{2}$).

Ultradensity conjecture. In the case of near half spaces, $\mathcal{V}$ should be ultradense in $S$.

Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain, and let the fractional Korn inequalities under consideration be the fractional analogues of Korn's first and second i…

Let $\Omega\subseteq\mathbb{R}^{d+1}$ be a domain with $0\in\partial\Omega$ whose boundary is $(\epsilon,d)$-Reifenberg flat. Theorem A asserts, under the additional hypothesis tha…

Let $\Omega$ be a two-dimensional bounded Lipschitz domain, and let $\mathrm{L}^p_{\sigma}(\Omega)$ denote the space of solenoidal vector fields in $\mathrm{L}^p(\Omega)$. Let $p$…

Let $\Omega$ be a bounded Lipschitz domain with boundary $\partial\Omega$, surface measure $\sigma$, and outward unit normal $\nu_{\mathrm{o}}$. The function $\nu_{\mathrm{o}}$ is…

Let $\Omega\subset\mathbb{R}^n$ be a Lipschitz domain and let $\Sigma\subset\partial\Omega$ be relatively open with respect to $\partial\Omega$. Let $u$ be harmonic in $\Omega$ and…

Positive Sobolev regularity conjecture. For every bounded Lipschitz domain $Omega$, there exists $varepsilon>0$ for which the displayed inclusion holds. The paper states that this…

Dual characterization conjecture. The duals are given by

Lipschitz-domain generalization conjecture. The corresponding layer-potential bounds should also hold when the half-space $\mathbb{R}^{n+1}_+$ is replaced by a Lipschitz graph doma…

Let $F_{\partial\Omega_{\pi-\theta}}(\alpha)$ denote the boundary functional associated with the domain $\Omega_{\pi-\theta}$, where $\theta\rightarrow 0+$, and let $c'>0$ be a con…

Let $\Omega$ be the Lipschitz domain under consideration, let $y\in\partial\Omega$, and let $u_{k\delta_y}$ denote the positive solution associated with boundary measure…