TWO-DIMENSIONAL MINIMAL GRAPHS OVER UNBOUNDED DOMAINS

Abstract

In this paper we will study solution pairs $(u,D)$ of the minimal surface equation defined over an unbounded domain $D$ in $R^2$, with $u=0$ on $\partial D$. It is well known that there are severe limitations on the geometry of $D$; for example $D$ cannot be contained in any proper wedge (angle less than $\pi$). Under the assumption of sublinear growth in a suitably strong sense, we show that if $u$ has order of growth $\alpha$ in the sense of complex variables, then the ‘asymptototic angle’ of $D$ must be at least $\pi/\alpha$. In particular, there are at most two such solution pairs defined over disjoint domains. If $\alpha<1$ then $u$ cannot change sign and there is no other disjoint solution pair. This result is sharp as can be seen by a suitable piece of Enneper’s surface which has order $\alpha=\tfrac{2}{3}$ and asymptotic angle $\tfrac{3}{2}\pi$.

AMS 2000 Mathematics subject classification: Primary 35J60; 53A10