Nonlocal Character of the Reduced Theory of Thin Films with Higher Order Perturbations

G. Dal Maso SISSA Via Beirut 2 34151 Trieste, Italy dalmaso@sissa.it

Irene Fonseca Carnegie Mellon University Department of Mathematical Sciences Pittsburgh, PA 15213 fonseca@andrew.cmu.edu

Giovanni Leoni Carnegie Mellon University Department of Mathematical Sciences Pittsburgh, PA 15213 giovanni@andrew.cmu.edu

Abstract: In this paper it is shown that, when there is lack of coercivity with respect to some partial derivatives on the underlying field u, then the relaxation of the functional

u ↦ ∫_{Ω} f(u, Du) dx

may fail to be local. This result is applied to a singular perturbation model for a membrane energy depending on deformations and out-of-plane bending.