CONDITIONS OF PARABOLIC AND HYPERBOLIC TYPES OF NONPARAMETRIC SURFACE

A nonparametric two-dimensional surface is considered, i.e. the graph of some smooth function 𝑓 = 𝑓(𝑥, 𝑦) defined over the entire plane. The surface type is defined in terms of the two-dimensional capacity of a compact set on the surface as follows. If the capacity of any non-degenerate continuum on the surface is zero, then the surface has a parabolic type, and if the capacity of any non-degenerate continuum on the surface is positive, then the surface has a hyperbolic type. The paper establishes integral conditions for the function 𝑓 defining the surface, under which the surface has one or another type. These conditions are expressed as conditions of convergence or divergence of the corresponding integrals and characterize the degree of growth (for ρ → +∞) of the partial derivatives |𝑓′ ρ | and |𝑓′ φ|, where ρ and φ are the polar coordinates of a point (𝑥, 𝑦) of the plane over which the surface lies. It turns out that if the degree of growth of the function |𝑓′ φ | is small, or more precisely, less than the growth of a linear function of ρ, then, regardless of the growth rate of |𝑓′ ρ |, the surface is of parabolic type. But if (for ρ → +∞) the growth of the function |𝑓′ φ| exceeds the linear growth of ρ, and at the same time exceeds the growth of |𝑓′ ρ|, then the surface is of hyperbolic type. Using the obtained conditions of the hyperbolic type of surface, the construction of an example of a nonparametric surface of hyperbolic type is described. This example complements the known examples of this type constructed by various famous mathematicians back in the late 50s of the last century. But the justifications of these examples, unlike the example proposed in this paper, do not rely (explicitly) on any general condition of the hyperbolic type of surface.