Distortion of the triangle isoperimetricity coefficient under quasiconformal mapping

D. V. Shurkaeva

Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 23 (2020) no. 1, pp. 22-31

Résumé

When solving problems of mathematical modeling on triangular and terahedral computational grids, it becomes necessary to estimate the error of the obtained solution, which depends on the degree of non-degeneracy of triangulation triangles. Therefore, long and narrow (“splinter”) triangles are avoided. We introduce the ratio $$ \sigma (\Delta) = \dfrac {{|\partial \Delta |}^{\frac {n}{n-1}}}{|\Delta|}, $$ called isoperimetricity coefficient of an $ n $-dimensional simplex $\Delta $. The value $ \sigma (\Delta) $ characterizes the deviation of an arbitrary simplex $ \Delta $ from the regular simplex, since the minimum value is reached on the regular simplex based on isoperimetric inequality. Let the mapping $ f: D \rightarrow D \; (D, D \subset \mathbb{R} ^ n) $ is homeomorphic and differentiable almost everywhere. Denoted by $ \lambda, \Lambda $ are the smallest and largest eigenvalues of the operator $ (d_{x_0} f)^ T (d_{x_0} f) $, respectively. For some interior point $ x_0 \in \Delta $ at which the mapping $ f $ is differentiable, we denote $$ B = B (x_0, f, \Delta) = \max_ {k = \overline {0, n}} \dfrac {| H_k |} {| P_k-x_0 |}, $$ where $$ H_k = H (x_0, P_k) = f (P_k) -f (x_0) -d_{x_0} f (P_k-x_0). $$ For a pair of simplex vertices $ P_i $ and $ P_j $, we introduce the notation $$ d_{ij} = | P_i-x_0 | + | P_j-x_0 |, \quad 0 \leqslant i <j \leqslant n. $$ Lemma. Let the domain $ D \subset \mathbb{R}^n $ and the simplex $ \Delta P_0P_1 ... P_n \subset D $ with the minimum and maximum edge lengths $ \rho_ {min} $ and $ \rho_ {max} $ respectively and the minimum face area is $S$, a homeomorphic and differentiable almost everywhere mapping $ f: D \rightarrow D '\; (D '\subset \mathbb {R} ^ n) $ and some interior point of the simplex $ x_0 \in \Delta $, in which the mapping $ f $ is differentiable with coefficient $ k = \dfrac{\sqrt {\Lambda} + B \tau }{\sqrt {\lambda} - B \tau }$. Then if the condition $ k <\sqrt [2n] {1+ \dfrac {2n \rho_ {min} ^ {2n-2} \rho_ {max} ^ 2- (n-1) \rho_ {min} ^ {2n}} { q_n \rho_ {max} ^ {2n}}} $ is satisfied, the ratio of the isoperimetricity coefficients of the image simplex and the inverse image simplex is estimated by the formula \begin{equation*} \dfrac{\sqrt[2(n-1)]{\left( {1-(k^{2(n-1)}-1) \theta_{n-1}}\right) ^n}}{k^n\sqrt{1+ \left(1-k^{-2n} \right) \theta_n}} \leqslant \dfrac{\sigma'}{\sigma} \leqslant k^n \dfrac{\sqrt[2(n-1)]{\left( {1+(1-k^{-2(n-1)}) \theta_{n-1}}\right) ^n}}{\sqrt{1- \left(k^{2n}-1 \right) \theta_n}}, \end{equation*} where $ \tau = \tau (\Delta, x_0) = \max \limits_ {0 \leqslant i <j \leqslant n} \dfrac {d_ {ij} } {\rho_ {ij}} $, $\theta_{n-1}=\dfrac{q_{n-1}\rho_{max}^{2(n-1)}}{r_{n-1}S^2}$, $\theta_n = \dfrac{q_n \rho_{max}^{2n}}{r_nV^2}$, $r_n=2^n(n!)^2$. In the case of quasiconformal mapping we obtain the following result. Theorem. Let $ D, \; D'$ are the regions of the complex plane $ \mathbb {C} $, triangle $ \Delta P_0P_1P_2 \subset D $ with side lengths $ a \geqslant b \geqslant c $ and the area of the triangle is $S$, and $ (\cdot) z_0 $ is the incenter of $ \Delta $, $ f: D \rightarrow D '$ is a differentiable quasiconformal mapping with the coefficient $ k = \dfrac {\| d_ {z_0} f \| _F ^ 2 \cdot \sqrt {1+ \mu} + \sqrt{2}B \tau} {\| d_ {z_0} f \| _F ^ 2 \cdot \sqrt {1- \mu} - \sqrt{2} B \tau} $. Then if $ k <\sqrt [4] {1+ \dfrac {4c ^ 2a ^ 2-c ^ 4} {3a ^ 4}}$, the ratio of the isoperimetricity coefficients of the image triangle and the inverse image triangle is estimated by the formula \begin{equation*} \dfrac{1}{k^2\sqrt{1+ \theta \left(1-k^{-4} \right)}} \leqslant \dfrac{\sigma’}{\sigma} \leqslant \dfrac{k^2 }{\sqrt{1- \theta (k^4-1)}}, \end{equation*} for $ \mu = \mu (f) = \sqrt {1- \dfrac {4 J_f ^ 2 (z_0)} {\| d_ {z_0} f \| ^ 4}} $, $ \tau = \tau (\Delta) = \dfrac {2 \sqrt {p}} {\sqrt {b}} \left (\dfrac {\sqrt {c (p-a)}} {a} + \dfrac {\sqrt {a (p-c)}} {c} \right) $, and $\theta = \dfrac{3 a^4}{16S^2}$, where $ p $ is the semiperimeter of the triangle, and $ J_f (z_0) $ is the Jacobian of the mapping $ f $ at the point $ z_0 $.

Détail

Keywords: triangle isoperimetricity coefficient, isoperimetric inequality, quasiconformal mapping, gomeomorphism, quasi-isometric mapping, Keli — Menger determinant.
Mots-clés : simplex, triangulation