Convexity criteria for expansions of the Landau-de Gennes elastic free energy

Figures & data

Figure 1. Contour plot of the region $\mathrm{ℛ}$ defined by Equation (17)(17) $$\begin{array}{cc}\mathrm{ℛ}\hfill & =\{(b,c,d):d\ge 0,\hspace{1em}c-\frac{1}{3}d>-1,\hspace{1em}\frac{5}{3}b+\frac{1}{6}c-\frac{1}{3}d>-1,\hspace{1em}\frac{1}{2}c+\frac{1}{3}d<1\}\hfill \\ \hfill & \hspace{1em}\cup \{(b,c,d):d<0,\hspace{1em}c+\frac{2}{3}d>-1,\hspace{1em}\frac{5}{3}b+\frac{1}{6}c+\frac{2}{3}d>-1,\hspace{1em}\frac{1}{2}c-\frac{2}{3}d<1\}.\hfill \end{array}$$(17) in the $({\mathcal{\ell }}_3,m_7)$-plane with ${\mathcal{\ell }}_2=\frac{L_2}{L_1},{\mathcal{\ell }}_3=\frac{L_3}{L_1}$ , and $m_7=\frac{M_7}{L_1}.$ When ${\mathcal{\ell }}_2>0$ the values of ${\mathcal{\ell }}_3$ and m₇ are bounded by the outer large irregular rectangle.

Figure 2. Contour plots of the region $\mathcal{D}$ defined by Equation (23)(23) $$\begin{array}{c}\mathcal{D}=\{({\kappa }_2,{\kappa }_3,{\kappa }_4):{\kappa }_3\ge 1,(\frac{1}{s}-1\left)\right({\kappa }_3-1)\le 3({\kappa }_2-{\kappa }_4),\\ (\frac{1}{s}-1)({\kappa }_3-1)\le 3({\kappa }_2+2{\kappa }_4),\\ (\frac{1}{s}-1)({\kappa }_3-1)\le 10-7{\kappa }_2-9{\kappa }_4\}\\ \cup \{({\kappa }_2,{\kappa }_3,{\kappa }_4):{\kappa }_3<1,(\frac{2}{s}+1\left)\right(1-{\kappa }_3)\le 3({\kappa }_2-{\kappa }_4),\\ \hspace{0.5em}(\frac{2}{s}+1)(1-{\kappa }_3)\le 3({\kappa }_2+2{\kappa }_4),\\ (\frac{2}{s}+1)(1-{\kappa }_3)\le 10-7{\kappa }_2-9{\kappa }_4\}.\end{array}$$(23) with $s=\frac{1}{2}$ in the $({\kappa }_2,{\kappa }_4)$-plane. The left-hand plot is for the case ${\kappa }_3\le 1$ while the right-hand plot is for the case ${\kappa }_3\ge 1.$