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

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.