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

TL;DR

This paper extends a classical theorem on the removability of isolated singularities from minimal surface equations to more general linear growth functionals, broadening the understanding of singularity behavior in nonlinear PDEs.

Contribution

It generalizes the Bers and Finn theorem to a wider class of Euler-Lagrange equations with linear growth, using generalized catenoids for comparison principles.

Findings

Removability of isolated singularities holds for general linear growth functionals.

Generalized catenoids are effective tools for proving comparison principles.

The results extend classical minimal surface theory to broader variational problems.

Abstract

A famous theorem of Bers and Finn states that isolated singularities of solutions to the non-parametric minimal surface equation are removable. We show that this result remains valid, if the area functional is replaced by a general functional of linear growth depending on the modulus of the gradient. We emphasize that Serrin ([1]) in fact proved the removability of singularities on sets of $(n-1)$-dimensional Hausdorff measure zero in an even more general setting. Our main interest is to generalize the comparison principles as outlined, for instance, in Section 10 of [2] without having the particular geometric structure of minimal surfaces. It turns out that generalized catenoids serve as an appropriate tool for proving our results.