Certain asymptotic properties of subsolutions of equations of minimal surface type (Q1080064)

Let G be a domain in \({\mathbb{R}}^ 2\) and let \(a_ i(x,\xi)\) be measurable functions defined for \(x=(x_ 1,x_ 2)\in G\) and \(\xi =(\xi_ 1,\xi_ 2)\in {\mathbb{R}}^ 2\). Let \(a=(a_ 1,a_ 2)\), \(| a| =(a^ 2_ 1+a^ 2_ 2)^{1/2}\), and suppose that there are constants \(v_ 1,v_ 2>0\) such that, for all admissible x and \(\xi\), the following two inequalities hold: \[ v_ 1| \xi |^ 2(1+| \xi |^ 2)^{-1/2}\leq \sum^{2}_{i=1}\xi_ ia_ i(x,\xi),\quad | a(x,\xi)| \leq v_ 2| \xi | (1+| \xi |^ 2)^{- 1/2}. \] Let \(\nabla f=(f_{x_ 1},f_{x_ 2})\) be the gradient of f and \(L(f)=\sum^{2}_{j-1}(d/dx_ j)a_ j(x,\nabla f)\). A locally Lipschitz function f in G is called a subsolution of \(L(g)=0\) if, for every locally Lipschitz test function \(\phi\geq 0\) with compact support in G, \(\int_{G}\sum^{2}_{i=1}\phi_{x_ i}a_ i(x,\nabla f)dx_ 1dx_ 2\leq 0.\) From now on assume that \(G={\mathbb{R}}^ 2\) and that f is a subsolution of \(L(g)=0\) in G and denote by \(O_ c\) a component of the set \(\{\) z: f(z)\(>c\}\). For \(c_ 1\leq c_ 2\leq...\leq c_ N\) suppose that there are unbounded pairwise disjoint sets \(O_{c_ j}\), \(j=1,...,N\). Denote by \(N_ f\) the largest number N with this property. The authors show that the order of f has a lower bound in terms of \(N_ f\) and the numbers \(v_ 1,v_ 2\). Some related results of this kind were proved by the second author [Mat. Sb., Nov. Ser. 111(153), No.1, 42- 66 (1980; Zbl 0428.35036)]. The main methods of proof include (a) the study of the moduli of some curve families on the surface \(z=f(x_ 1,x_ 2)\), (b) ideas from the theory of quasiconformal mappings, and (c) appropriate choice of the test function \(\phi\). The second author applied a similar method earlier [ibid. 108(150), No.2, 268-289 (1979; Zbl 0488.49029)].