Some estimates of the asymptotic behavior of the minimal surface over strip domain

R. S. Akopyan

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 3 (2014), pp. 6-12

Résumé

The solutions of equation of the minimal surfaces given over unbounded domains were studied in many works (for example, see [1–3; 5]) dealing with various problems of asymptotic behavior of the minimal surfaces, including the questions of admissible speed of stabilization and the theorem by Fragmen — Lindelef. The object of the present research is solution of equations of the minimal surfaces given over strip domains of special type and satisfying some zero boundary values. The author estimates the possible asymptotic behavior of Gaussian curvature using the traditional for such kind of problems approach consisting in construction of auxiliary conformal mapping, the appropriate properties of which are investigated. Two special cases are studied. Let $z=f(x,y)$ be the $C^2$-solution of the equation of minimal surfaces (1) given over strip domain $\Pi = \{(x,y)\in R^2: 0<x<+\infty ,\; -\varphi(x)<y<\varphi(x) \},$ where $\varphi(x)$ — continuously differentiable function. Let us denote by the symbols $\partial ' \Pi$ and $\partial '' \Pi$ sectors of the boundary $\partial \Pi$: $$ \partial ' \Pi = \partial \Pi \cap \{(x,y) \in R^2: x=0\},\quad \partial '' \Pi = \partial \Pi \setminus \partial ' \Pi . $$ Assume that the solution $z=f(x,y)$ satisfies the conditions (i) and (ii). For the Gaussian curvature of minimal surfaces $K(x,y)$ the following theorems are suggested: Theorem 1. Let $\nu(x)$ — positive, non-decreasing continuous on $(0,+\infty)$ the function to which $$ \int\limits_0^{+\infty}\nu(x)e^{-\sigma(x)}\frac{\ dx}{\mu(x)} = +\infty, \qquad {\rm where \ \ } \sigma(x) = \pi \int\limits_0^{x}\frac{\ dt}{\mu(t)}. $$ Then, if everywhere in $\Pi$ executed $$ {\rm log}{(-K(x,y))}\leq -\nu(x), $$ then $f(x,y)\equiv const$. Let $\lambda(x) = \pi \int\limits_0^{x}\frac{1+{\frac{1}{12}}{\mu'}^2(t)}{\mu(t)}\ dt.$ Theorem 2. Let $L$ — curve starting at any endpoint of the border domain $\Pi$ and goes to infinity, remaining in $\Pi$. If $K(x,y)$ is bounded in $\overline {\Pi}$, and $$ \frac{\log(-K(x,y))}{e^{\lambda(x)}}\to -\infty, \qquad (x,y)\in L,\qquad x\to +\infty, $$ then $f(x,y)\equiv const$. Similar results on the speed of approach to zero of Gaussian curvature the minimal surface were obtained in [1; 2]. However, in the considered special cases at the greater community, they are less exact.

Détail

Keywords: equations of the minimal surfaces, gaussian curvature, asymptotic behavior, holomorphic functions.
Mots-clés : strip domain

Bibliographie

[1] R.\;S. Akopyan, “On the Admissible Speed of Approaching to Zero of Gaussian Curvature of Minimal Surface Over Strip Domain”, Science Journal of Volgograd State University. Mathematics. Physics, 2012, no. 2 (17), 4–8

[2] R.\;S. Akopyan, “Theorems Fragmen–Lindelef Type for the Minimal Surface Over Strip Domain”, Science Journal of Volgograd State University. Mathematics. Physics, 2013, no. 2 (19), 6–12

[3] V.\;M. Miklyukov, “Some Issues of Quality Theory on the Equations of Minimal Surface Type”, Boundary value problems of mathematical physics, Naukova dumka Publ., Kiev, 1983, 137–146

[4] R. Oserman, “Minimal surfaces”, Russian Mathematical Surveys, XXII:4 (1967), 55–136

[5] V.\;I. Pelikh, “Theorems of Fragmen–Lindelef on Minimal Surfaces”, Geometrical Analysis and Its Applications: Scientific Schools of VolSU, 1, 1999, 352–368