Fragm'en - Lindel"of type theorems for the minimal surface over strip domain

Estimations of possible asymptotic behavior of Gaussian curvature the minimal surfaces given over strip domain are received in this paper. To research of solutions of equation of the minimal surfaces given over unbounded domains, many works (see, for example, [1-3; 5; 6; 8-11]) in which various tasks of asymptotic behavior of the minimal surfaces were studied, including questions of admissible speed of stabilization and theorem by Fragmen - Lindelef are devoted. As our object of research there are solutions of equation of the minimal surfaces given over strip domain and satisfying some zero boundary values. We use a traditional approach for the solution of a similar kind of tasks consisting in construction of auxiliary conformal mapping which appropriate properties are studied. Let z = f (x, y) be the C 2 - solution of the equation of minimal surfaces (1) given over strip domain П = {( x, y ) є R 2:0 1( x) 2( x)} where φ 1 (x) = φ(x) — ½θ(x), φ 2(x) = φ(x) +½θ(x), φ(x),θ(x) — continuously differentiable when x > 0 functions, satisfying the conditions: θ(x) > 0, lim φ(x) = lim θ'(x) = 0. x→+∞ x→+∞ Let us denote by the symbols ∂'П and ∂"П sectors of the boundary ∂П : ∂'П = ∂П I {(x, у) є R 2 : x = 0} ∂''П = ∂П \ ∂'П. Assume that the solution f (x, у) є C 1 (П) and satisfies the condition (2). The following theorem is valid for the Gaussian curvature of minimal surfaces K( x, y). Let N ( x) is determined by the formula (6). Theorem. Let L — curve starting at any endpoint of the border domain П and going to infinity, remains in П. If К(x, y) is bounded in П, and log(-K ( x, y)) — ------ →∞, x — +∞, (x, у) є L, N ( x) then f ( x, y) is a linear function. Similar results of the speed of approach to zero of Gaussian curvature considered above the minimal surface were obtained in [1]. The special example of minimal surfaces, defined over a semistrip, was presented in [2; 3].