Thin elastic films: The impact of higher order perturbations

Abstract

Abstract. The asymptotic behavior of an elastic thin film penalized by a van der Wals type interfacial energy is investigated when both its thickness and the magnitude of the additional energy vanish in the limit. Keeping track of both mid-plane and outof-plane deformations (through the introduction of the Cosserat vector), the resulting behavior strongly depends upon the ratio between thickness and interfacial energy.

Erratum to “Thin elastic films: the impact of higher order perturbations”

“…The proof is very similar to that of Theorem 4.2 in [2], and thus we only indicate the main changes.…”

“…Remark E. A proof almost identical to that of Remark 4.3 in [2] shows that W 2 (·, ·) is upper semi-continuous on R 2×3 × W 1,2 (I; R 3 ) equipped with its strong topology. We now turn to the proof of Theorem B.…”

“…The lower bound in the original Theorem 3.1 in [2] is correct. The mistake in the proof of Theorem 3.1 was in trying to show that the upper bound is still H in Ω (a condition that is automatically satisfied when γ < 2 in the finite energy regime), then indeed…”

“…We also introduce H γ 2,3 and H 2 2,3 , which we define analogously except for the fact that the approximating sequences {u ε } belong to W 1,q (ω; R 3 ). The proof of Theorem 3.1 in [2] has an error in the upper bound. To date this upper bound can only be established if γ < 2 or if D 3 b = 0 L 3 a.e.…”

“…To see this, we refer the reader to the proof of Theorem 4.1 in [2], which still works unchanged for any γ > 0 when D 3 b = 0 L 3 a.e. in Ω.…”