Second-order energies on thin structures: variational theory and non-local effects

For a given positive measure μ on Rn, we consider integral functionals of the kindF(u)=∫Rnf(x,∇u,∇2u) dμ, u∈C0∞(Rn),and we study their relaxation with respect to the Lμp topology, p being the growth exponent of f. To obtain the relaxed energy F, we develop a suitable theory of second-order μ-intrinsic operators, related to a Cosserat vector field and to a curvature tensor. Our main theorem shows that the functional F is in general a non-local one; this unexpected feature occurs even in very simple examples, when μ is the one-dimensional Hausdorff measure over a closed Lipschitz curve in the plane