We give an example of an autonomous functional $F(u) = \int_\Omega f(u,Du) dx$ (where $\Omega$ is open subset of $R^2$ and $u:\Omega\to R^2$ belongs the Sobolev space $W^{1,1}$) which is sequentially weakly lower semicontinuous in $W^{1,p}$ for every $p \ge 1$ 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.