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

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.


Introduction
We discuss solutions u ∈ among functions v: Ω → R with prescribed boundary data. The assumptions concerning the density g are as follows: we consider functions g: [0, ∞) → R of class C 2,α [0, ∞) for some exponent 0 < α < 1 being of linear growth in the sense that with suitable constants a, A > 0, b, B ≥ 0 the inequality holds for any t ≥ 0. Moreover, we require strict convexity of g by imposing the condition g ′′ (t) > 0 for all t ≥ 0 . (1.4) Finally, we assume g ′ (0) = 0 . (1.5) We then will prove that the famous theorem of Bers and Finn (see [3], [4]) on the removability of isolated singularities for solutions of the non-parametric minimal surface equation extends to any solution of (1.1) provided that g satisfies these hypotheses.
In more detail we have the following result: Then u admits an extension u ∈ C 2 (Ω) and u solves equation (1.1) on the set Ω.
In the case of minimal surfaces, i.e. for the choice g(t) = √ 1 + t 2 in equation (1.1) and for n = 2, the result of the theorem was proved independently by Bers [3] and Finn [4]. Concerning solutions of the non-parametric minimal surface equation in dimensions n > 2 the removability of singular sets K being closed subsets of Ω such that H n−1 (K) = 0 was established by DeGiorgi and Stampacchia [5], by Simon [6], Anzellotti [7] and Miranda [8].
During the proof of Theorem 1 we will have to distinguish two essentially different cases, where the first one is closely related to the minimal surface setting in the sense that we suppose restricting the growth of g ′′ at infinity. Note that (1.6) is a consequence of the pointwise inequality provided we choose µ > 2. In the minimal surface case, i.e. 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 [9] 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.
However, this removability property does not depend on any geometric features. As it is formulated in Theorem 1, the non-existence of isolated singularities is just a consequence of the linear growth of g which is also exploited in the second case This condition already occurs, e.g., in [10] (compare also [11]) in a quite different setting: in Theorem 1.1 of [10], equation (1.8) together with some kind of balancing condition serves as a criterion for the solvability of a classical Dirichlet-problem, where the authors argue with the help of suitable barrier functions. Both in [10] and in [11] generalized catenoids are used as basic tools, which is also the case in our considerations. Depending on the conditions (1.6) and (1.8), respectively, these catenoids are of infinte height or uniformly bounded.

Proof of Theorem 1 under condition (1.6)
In the following we consider energy densities g: [0, ∞) → R of class C 2,α such that (1.3)-(1.5) hold. In particular g ′ is a bounded function and strictly increasing, thus Step 1. Maximum principle.
We observe that in the subsequent considerations we may not assume Lipschitz continuity of solutions up to the boundary, hence Theorem 1.2 of [12] does not apply. We will make use of the following variant: Suppose that D is a bounded Lipschitz domain in R n and that we have (1.3)-(1.5). Moreover suppose that u, v ∈ C 2 (D) ∩ C 0 (D) satisfy equation (1.1). Then we have: With Lemma 1 the following corollary is immediate: Now we refer to Theorem 10.1, p. 263, of [13] with coefficients (x ∈ D, y ∈ R, which, by (2.3), are seen to be elliptic. Since we consider the admissible function space C 2 (D) ∩ C 0 (D) the proof is complete with the above mentioned reference.
Proof of Lemma 2. W.l.o.g. we let α = 1, a = 0 in the definition of k ± α,a and l ± α,a , respectively. Dropping the indices α and a we have for t > 1 (letting is well defined at least for t > 1 on account of assumption (1.6) and due to the behaviour of g ′ as stated in (2.1).
Moreover, from (2.6) it immediately follows that and obviously l + solves (2.7) for the choices r = 1, R = ∞. This proves Lemma 2, since with obvious modifications the above calculations can be adjusted to the functions k − α,a .
Here a ∈ R is chosen according to (note k ± r,a (x) = a for |x| = 1) Since u is bounded on ∂B r 1/(n−1) (0) and since we have (3.2), we may choose ε > 0 sufficiently small such that on ∂B (r+ε) 1/(n−1) (0) . where the constant c is not depending on x, hence we have established an uniform upper bound for u and a uniform lower bound follows along similar lines. Proceeding exactly as done in the first case at the end of Step 4 the theorem is proved.