15 problems

Unique stable BD profile conjecture. For $K>4$, the $\mathbf{P}^b$-$BD$-$\mathbf{P}^t$ configuration is the unique stable mixed three-dimensional critical point on $V_K$.

Meta-state minimiser conjecture. For odd $K$, the $Meta$ states are the $z$-invariant energy minimisers, with the splay vertices furthest apart.

BD minimiser conjecture. The $\mathbf{P}^b$-$BD$-$\mathbf{P}^t$ configuration is the unique reduced Landau-de Gennes minimiser for small $h$ and remains potentially stable for all…

Index comparison conjecture. The index of $p_u$ is always higher than the index of the mixed three-dimensional critical point with a $p_u$-type interior profile.

Planarity conjecture. After the field is removed, the director remains in the plane of the polygon.

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.

Let $S_1,\ldots,S_4$ denote the four short edges of the truncated square, and let ${\bf Q}_{\mathrm{b}}$ be the prescribed Dirichlet boundary value on these edges. Although these e…

Let the square have edge length $\lambda$, with its vertices truncated so that the domain has four long edges inherited from the square and four artificial short edges connecting t…

Competing-map conjecture. There are at least two competing limiting harmonic maps for this two-dimensional problem, namely

Let $c$ be the Landau–de Gennes tensor order parameter at a point on a disclination line, and let $s$ denote its scalar order parameter and $c{n}$ its director. Disclination-line c…

Consider disclination lines in the Landau–de Gennes theory, including $7k/2$-disclination lines and the $71/2$-disclination line. Half-integer disclination-line stability conjectur…

Consider defects in the Landau–de Gennes model, including $k$-disclination lines whose profiles are obtained from the corresponding profile equations. Profile conjecture. The local…

In the Landau–de Gennes model, let point defects and disclination lines be the two defect forms, with excessive energy concentrated near the defects. Energy-concentration genericit…

Let $W_\beta$ be the generalized potential considered in the paper, let $S$ be the relevant set of admissible states, and let the conclusion of the projection lemma be the inequali…

Let $u_0$ be the limiting map in Theorem 1 and let $\psi_0$ be the function associated with $u_0$ there, satisfying the stated matrix-valued elliptic equation and boundary conditio…