Liouville-type results in two dimensions for stationary points of functionals with linear growth

Abstract

1 We consider variational integrals of linear growth satisfying the condition of µ-ellipticity for some exponent µ > 1 and prove that stationary points u: R 2 → R N with the property lim supmust be affine functions.

Small weights in Caccioppoli's inequality and applications to Liouville-type theorems for non-standard problems

“…As outlined in [6] (compare the discussion after inequality (1.3) in this reference), Assumption 4.1 implies with constants a, A > 0, b, B ≥ 0 and for all Z ∈ R 2 the linear growth condition…”

“…Let us give some further explanations concerning the hypotheses of Theorem 1.1: from (1.2) and the boundedness of ∇f it follows that f is of linear growth, and this actually holds for arbitrary exponents µ ≤ 1 < µ. We refer to Lemma 2.1 in [6].…”

“…Before summarizing some corollaries of Theorem 1.2 in this particular setting, we will show the following proposition which follows the line of the proof of Theorem 1.1 of [6].…”

“…Condition (1.2) also serves as one main assumption in the recent paper [6] on Liouville-type results in two dimensions for functionals satisfying a linear growth condition. These results are restricted to the case µ = 1, for instance we have:…”

“…Contributions in the case of linear or anisotropic growth are quite rarely found. We just mention the papers [3], [4], [5], [6] and the references quoted therein.…”