Mathematics and mechanics

Distortion of the triangle isoperimetricity coefficient under quasiconformal mapping

D. V. Shurkaeva

National Research University "Moscow Power Engineering Institute"

Abstract: 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 $.

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

UDC: 519.65
BBK: 22.161.5

Received: 10.10.2019

DOI: 10.15688/mpcm.jvolsu.2020.1.3