33 problems

Strict positivity conjecture. For every $b>0$, the second variation at $Y$ is strictly positive definite.

H-theorem conjecture. A form of the H-theorem remains valid even when $mathcal{A}$ is not transitive.

Conjecture. Recovering the complete Leslie viscous stress tensor is possible with an anisotropic BGK collision operator involving several relaxation timescales.

Degond–Frouvelle–Liu conjecture. The only stable anisotropic equilibria occur when $k=1$ and $eta>eta_1^*$; consequently, every stable anisotropic equilibrium is axisymmetric. For…

Let $V$ be the cuboid under consideration, and let the BD-BD$_2$-BD$_2$ critical point be a critical point of the total energy whose profile is uniaxial near the center of $V$ and…

Let $\Omega$ be the computational domain, let $\vectorsym S$ denote the nematic liquid-crystal tensor field, and let $\vectorsym f$ be the fiber field. The forcing term is the scal…

Aviles–Giga conjecture. The $\Gamma$-limit of $AG_{\varepsilon}$ as $\varepsilon\to0$ is a functional finite on such functions and charges jumps of the gradient field…

Let $n\geq 2$, let $\rho^n$ denote the function associated with equilibria whose largest eigenspace has dimension $d=1$, and let $\rho^*>0$ be the threshold density. For…

Non-axisymmetric bifurcation conjecture. For every $r>1$, the bifurcation from this eigenvalue is to non-axisymmetric critical points.

Let a solution on the hexagonal domain have bend-like vertices, meaning vertices where an associated defect splits from the vertex into an interior defect and a defect at the verte…

Differential 1-form representation conjecture. In the continuous limit, a smectic is faithfully represented by a differential 1-form.

Universality conjecture. For arbitrary $A<0$, some qualitative properties of these critical-point solutions are universal, even though their $q_3$ profiles are non-constant.

Uniqueness and symmetry conjecture. There exists a unique minimizer of $\mathcal F_{\varepsilon}$ in $H^1_{Q_b}(D,\mathcal S_0)$, and this minimizer is $\mathbb Z_2\times O(2)$-sym…

Let the three-dimensional nematic liquid-crystal problem be the problem obtained by extending the thin-film problem under discussion to a three-dimensional geometry, with the direc…

Let $d\in\{2,3\}$ and let $\alpha>d/2$. Consider the stochastic Ericksen–Leslie equations with initial data…

Large-aspect-ratio minimizer conjecture. For $L/H \geqslant L_1$, there exists a two-dimensional minimizer $u_L$ such that $E_0[u_L]$ is lower than the minimum energy achieved over…

Intermediate-regime minimizer conjecture. For $L/Hin (L_0,L_1)$, the critical point constructed in the proof of Theorem textnormal{not1d} is a minimizer of $E_0$.

Small-aspect-ratio minimizer conjecture. For $0 < L/H < L_0$, the one-dimensional minimizer from Theorem 1 is a unique minimizer of $E_0$ among all two-dimensional competitors.

A nematic liquid crystal is described by a tensorial order parameter $Q$, known as a Q-tensor, rather than by a director field. Consider nematic point defects in the vicinity of mo…

Consider a monomer-dimer system with a strong attractive dimer-dimer interaction that favors alignment. A nematic liquid crystal phase is a phase in which the dimers are mostly ali…

The energy functional $E$ admits global minimizers with localized vortices, and let their rescaled local profile be the profile near such a vortex. Generalized second Painlevé prof…

Non-coercivity conjecture. The functional $\hat{\mycal E}_\infty$ is not coercive on $\hat X_\infty$.

Molecular symmetry transmission conjecture. For every local minimum $f$ of each quadratic kernel determined by the symmetries $D_{\infty h}$, $C_{\infty v}$, $D_{2h}$, and $C_{2v}$…

The theorem considers $1<r<2$ and assumes $(u_0,\nabla d_0)\in \dot{B}_{\infty,2}^{-1}$ in order to obtain $(u,\nabla d)\in L^2_tL^\infty_x$. Restriction conjecture. The restrictio…

Discontinuity-set conjecture. One has