Absence of Lavrentiev Phenomenon for Functionals with (p,q)-growth

The aim of this thesis is to provide the absence of Lavrentiev phenomenon for integral functionals of the following type • F(u) = \int_Ω f(x,Du(x)) dx, where Ω is a subset of ℝⁿ and x → ∂f/∂z is α-Hölder continuous, we denote f=f(x,z). Moreover, the density f is convex and satisfies the (p,q)-growth condition • |z|^p ≤ f(x, z) ≤ L(1 + |z|^q), with • 1 < p < q < p + pα/n. For the model density represented by the double phase functional • f(x,z) = |z|^p + a(x)|z|^q, we can do better, we can replace the relation between p and q written above with the following • 1 < p < q < p + k, where k ∈ (0,∞), provided • a(x) ≤ c[ a(y) + |x-y|ᵏ ].