Gap Phenomenon for Some Autonomous Functionals

We give an example of an autonomous functional $F(u) = \int_\Omega f(u, Du)dx$ (Ω open subset of ℝ² , u : Ω → ℝ² in the Sobolev space $W^{1,1}$ ) which is sequentially weakly lower semicontinuous in $W^{1,p}$ for every p ≥ 1 but does not agree with the relaxation of the same functional restricted to smooth functions when p < 2. A Lavrentiev phenomenon occurs for a related boundary problem.