Variants of Bernstein’s theorem for variational integrals with linear and nearly linear growth

1 Using a Caccioppoli-type inequality involving negative exponents for a directional weight we establish variants of Bernstein’s theorem for variational integrals with linear and nearly linear growth. We give some mild conditions for entire solutions of the equation


Introduction
In [1] Bernstein proved that every C 2 -solution u = u(x) = u(x 1 , x 2 ) of the non-parametric minimal surface equation div ∇u 1 + |∇u| 2  = 0 (1.1) over the entire plane must be an affine function, which means that with real numbers a, b, c it holds For a detailed discussion of this classical result the interested reader is referred for instance to [2], [3], [4], [5] and the references quoted therein.
Starting from Bernstein's result the question arises to which classes of second order equations the Bernstein-property extends.More precisely, we replace (1.1) through the equation for a second order elliptic operator L and assume that u ∈ C 2 (R 2 ) is an entire solution of (1.2) asking if u is affine.To our knowledge a complete answer to this problem is open, however we have the explicit "Nitsche-criterion" established by J.C.C. Nitsche and J.A. Nitsche [6].
In our note we discuss equation (1.2) assuming that L is the Euler-operator associated to the variational integral with density f : R 2 → R and for domains Ω ⊂ R 2 , i.e. (1.2) is replaced by div Df (∇u) = 0 , ( where in the minimal surface case (1.1) we have being an integrand of linear growth with repect to ∇u.For this particular class of energy densities and under the additional assumption that f is of type with exponent µ ≥ 3 (including the minimal surface case) we proved the Bernstein-property in Theorem 1.2 of [7] benefiting from the work [8] of Farina, Sciunzi and Valdinoci.
Bernstein-type theorems under natural additional conditions to be imposed on the entire solutions of the Euler-equations for splitting-type variational integrals of linear growth have been established in the recent paper [9].
One of the main tools used in [9] is a Caccioppoli-inequality involving negative exponents which was already exploited in different variants in the papers [10], [11], [12], [13].
In fact we used this inequality to show that ∂ 1 ∇u ≡ 0 which follows by considering the bilinear form D 2 f (∇u) with suitable weights.Since we are in two dimensions, the use of equation (1.3) then completes the proof of the splitting-type results in [9].
In the manuscript at hand we observe that even without splitting-structure it is possible to discuss the Caccioppoli-inequality with a directional weight obtaining ∂ 1 ∇u ≡ 0 and to argue similar as before.We note that considering directional weights gives much more flexibility in choosing exponents than arguing with a full gradient (see Proposition 6.1 and Proposition 6.2 of [13]).We here already note that we also include a logarithmic variant of the Caccioppli-inequality as an approach to the limit case α = −1/2.
Before going into these details let us have a brief look at power growth energy densities as for example with exponent s > 1.Then the Nitsche-criterion (compare [6], Satz) shows the existence of non-affine entire solutions to equation (1.3), and as we will shortly discuss in the Appendix the same reasoning applies to the nearly linear growth model which means that we do not have the Bernstein-property for equation (1.3) with density of the form (1.4).
However, as indicated above, we can establish a mild condition under which any entire solution is an affine function being valid for a large class of densities f including the nearly linear and even the linear case.

Let us formulate our
Assumptions.The density For a constant λ > 0 it holds with some constant c > 0.
We have the following result: Theorem 1.1.Let f satisfy (1.5) and (1.6) and consider an entire solution u ∈ C 2 (R 2 ) of equation (1.3).Suppose that with numbers 0 ≤ m < 1, K > 0 the solution satisfies Then u is affine.
Remark 1.2.Since the density f from (1.4) fulfills the the conditions (1.5) and (1.6) and since in this case non-affine entire solutions exist, the requirements (1.7) and (1.8) single out a class of entire solutions of Bernstein-type.
Of course we know nothing concerning the optimality of (1.7) and (1.8).
Another unsolved problem is the question, if in the case of linear growth with radial structure, i.e. f (∇u) = g(|∇u|), Bernstein's theorem holds without extra conditions on the entire solution u.
Remark 1.3.The conditions (1.7) and (1.8) are in some sense related to the "balancing conditions" used in [9] in order to exclude entire solutions of the form u(x 1 , x 2 ) = x 1 x 2 for densities f of splitting type.
Let us pass to the linear growth case replacing (1.6) by with a positive constant Λ.Here the notion of linear growth just expresses the fact that from (1.9) it follows that with some number c > 0. In this situation we have Theorem 1.2.Let f satisfy (1.5) together with (1.9) and let u ∈ C 2 (R 2 ) denote an entire solution of equation (1.3) for which we have with some number K ∈ (0, ∞).Then u is an affine function.
The results of Theorem 1.1 and Theorem 1.2 are not limited to the particular coordinate directions e 1 = (1, 0) and e 2 = (0, 1), more precisely it holds: Theorem 1.3.Let f satisfy either the assumptions of Theorem 1.1 ("case 1") or of Theorem 1.2 ("case 2") and suppose that u ∈ C 2 (R 2 ) is an entire solution of (1.3).Assume that there exist two linearly independent vectors Then u is an affine function.
The proof of this result follows from the observation that the function u is a local minimizer of the energy f (∇u) dx combined with a suitable linear transformation.If we let then it holds and ũ is an entire solution of equation (1.3) with f being replaced by f , which follows from the local minimality of ũ with respect to the energy f (∇w) dx.
Obviously the properties of f required in Theorem 1.1 and Theorem 1.2, respectively, are consequences of the corresponding assumptions imposed on f , thus we can apply our previous results to ũ (and f ).
Our paper is organized as follows: in Section 2 we present the proof of Theorem 1.1 based on a Caccioppoli-inequality involving negative exponents, which has been established, for instance, in [13], Proposition 6.1.
Section 3 is devoted to the discussion of Theorem 1.2.We will make use of some kind of a limit version of Caccioppoli's inequality, whose proof will be presented below.With the help of this result the claim of Theorem 1.2 follows along the lines of Section 2. We finish Section 3 by presenting a technical extension of Theorem 1.2, which just follows from an inspection of the arguments (compare Theorem 3.1).
For the reader's convenience we discuss in an appendix equation (1.3) for the nearly linear growth case (1.4) and show that the Nitsche-criterion applies yielding non-affine solutions defined on the whole plane.
2 Proof of Theorem 1.1 Let f satisfy (1.5) and (1.6), let u denote an entire solution of (1.3) and assume w.l.o.g. that (1.7) holds.We apply inequality (107) from Proposition 6.1 in [13] with the choices l = 1, i = 1 and with a finite constant independent of R.
Letting η = 1 on B R and assuming |∇η| ≤ c/R we apply (1.6) to the r.h.s. of (2.1) and get On account of (1.7) we deduce for any ε > 0 Recall that m < 1, hence 2m(1 + α) − 1 < 0 for α > −1/2 sufficiently close to −1/2.We fix α with this property and finally select ε > 0 such that 2m(1 + α) − 1 + ε ≤ 0 to obtain We quote equation ( 108) from [13] again with the previous choices l = 1, ), 0 ≤ η ≤ 1, and with α as fixed above.The same calculations as carried out after (108) then yield On the r.h.s. of (2.5) we apply the Cauchy-Schwarz inequality to the bilinear form D 2 f (∇u), hence l.h.s. of (2.5) Here η has been chosen in such a way that η ≡ 1 on B R and therefore spt(∇η) ⊂ B 2R − B R .By (2.4) we have lim whereas the calculations carried out after (2.2) imply the boundedness of T 2 (R).Thus (2.5) and (2.6) imply hence ∇∂ 1 u = 0 on account of (1.5).This shows ∂ 1 u = a for some number a ∈ R and since for a constant c.Finally we observe that the function is strictly increasing (recall (1.5)), which shows the constancy of ϕ ′ and therefore u(0, x 2 ) = bx 2 + c for some numbers b, c ∈ R. Altogether we have shown that u is affine finishing the proof of Theorem 1.1.
3 Proof of Theorem 1.2 Let the assumptions of Theorem 1.2 hold and consider an entire solution u ∈ C 2 (R 2 ) of (1.3) without requiring (1.10) or (1.11) for the moment.If we use condition (1.9) in inequality (2.1) and if we assume that the choice α = −1/2 is admissible in (2.1), then the calculations of Section 2 would immediately imply that ∇ 2 u = 0 yielding Bernstein's theorem, i.e. the entire solution u is an affine function without adding further hypotheses on u.
However, we do not have (2.1) in the case that α = −1/2 and hence we provide a weaker version involving conditions like (1.10) or (1.11) in order to conclude that u is affine.
With ψ from above and Φ defined according to (3.1) we obtain Using the identity for some constant c > 0. Altogether we deduce from (3.3) the inequality of (3.7) is bounded by (recall (1.9)) Quoting (1.10) and returning to (3.7) we find that ) together with (1.9) and choose ρ according to (3.9).Suppose that u ∈ C 2 (R 2 ) is an entire solution of (1.3) such that holds with some finite constant c.Then u is an affine function.
We leave the details to the reader just adding the obvious remark that clearly we can interchange the roles of the partial derivatives ∂ 1 u and ∂ 2 u or even work with arbitrary directional derivatives as done in Theorem 1.3.