15 problems

Let $n\geq 2$, let $g(y',s)=\psi(y')\phi(s)\mathbf e_n$ be the boundary data for the Stokes system in the half space, where $\psi\geq 0$, $\psi\in C^\infty_c(B'_{1/2})$,…

Higher-dimensional extension of the auxiliary elliptic estimate. The estimate

Stability conjecture. The stability property extends to three dimensions and to domains that are not topologically trivial.

Let $h_1$ and $h_2$ describe the profiles of two closely spaced rigid inclusions, with the symmetric case given by $h_1(x_1)=h_2(x_1)$. Asymmetry conjecture. The asymmetry … could…

The setting involves the projection $P{ abla Qh}$ onto the discrete gradient space associated with $Qh$, whose $L^q$-stability is used in improved error estimates for the…

Non-unit eigenvalue count conjecture. The condition number of the Schur complement matrix is influenced by the number of non-unit eigenvalues in its spectrum.

Crouzeix–Falk's conjecture. The pair

Let $\Omega$ be convex, let $\mathcal{T}_h$ satisfy a maximum angle condition, and let the ${\mathbf{H}}(\operatorname{div})$ interpolation operator be…

Normal-continuity conjecture. The improved accuracy for $m=k$ is due to a better enforcement of continuity of the normal component of the velocity.

All-$p$ resolvent estimate conjecture. The resolvent estimate should be valid for all $1<p<\infty$ if $\Omega$ is convex.

Let $\undertilde{G}{}^3_{h0}$ and $\mathbb{P}^2_{h0}$ denote the cubic velocity and quadratic pressure spaces in the discretized Stokes pair on a triangulation. Assume that the tri…

Let $A^*$ be the homogenized Stokes tensor and let the propagation relation associate Bloch-wave data with $A^*$. Here $I$ denotes the $d\times d$ identity matrix and $c$ is a cons…

Let $f\in L^2(\Omega;\mathbb{R}^2)$, where $\Omega$ is a polygonal Lipschitz domain, and let $(u,p)\in H^1_0(\Omega;\mathbb{R}^2)\times L^2_0(\Omega)$ solve the Stokes equations. C…

Let $Q_T=\Omega\times(0,T)$, where $\Omega$ is the spatial domain, and let $v$ be a divergent-free function satisfying … Then…

Xie's conjecture. The optimal constant should be