Some geometric properties of nonparametric $μ$-surfaces in $\mathbb{R}^3$

Abstract

1 Smooth solutions of the equation div g ′ |∇u| |∇u| ∇u = 0 are considered generating nonparametric µ-surfaces in R 3 , whenever g is a function of linear growth satisfying in addition ∞ 0 sg ′′ (s)ds < ∞ .Particular examples are µ-elliptic energy densities g with exponent µ > 2 (see [1]) and the minimal surfaces belong to the class of 3surfaces.Generalizing the minimal surface case we prove the closedness of a suitable differential form N ∧dX. As a corollary we find an asymptotic conformal parametrization generated by this differential form.

An extension of a theorem of Bers and Finn on the removability of isolated singularities to the Euler-Lagrange equations related to general linear growth problems

“…for the choice g(t) = √ 1 + t 2 , we can choose µ = 3 in estimate (1.7), and by a "µ-surface in R n+1 " we denote the graph x, u(x) ∈ R n+1 : x ∈ D of a solution u: D → R of equation (1.1), provided that g satisfies the conditions (1.3), (1.4) and (1.7) for some exponent µ > 2. We refer to the recent manuscript [7] on some geometric properties of µ-surfaces in the case n = 2. Adopting this notation we deduce from Theorem 1 that µ-surfaces do not admit isolated singular points.…”