On a~new approach to Bernstein's theorem and related questions for equations of minimal surface type

V. M. Miklyukov

Sbornik. Mathematics, Tome 36 (1980) no. 2, pp. 251-271

Résumé

Differential operators $\mathscr L$ are considered on a surface $(D,ds^2)$ defined over a domain $D\subset\mathbf R^2$ by a line element $ds^2$ and having parabolic conformal type. Under certain conditions on $\mathscr L$, expressed in terms of the metric of the surface, three theorems on the solutions or subsolutions of the differential equation $\mathscr L[\varphi]=0$ are proved. These are Liouville's theorem, the Phragmén–Lindelöf theorem and a theorem on the behavior of solutions in a neighborhood of the point at infinity. By varying the choice of the metric $ds^2$, the corresponding results both for uniformly elliptic equations and for nonuniformly elliptic equations are obtained as corollaries of these general statements. Thus, for example, direct corollaries of the first theorem are Liouville's theorem for harmonic functions and Bernstein's theorem for minimal surfaces. Bibliography: 14 titles.