*Published Paper*

**Inserted:** 22 dec 2001

**Last Updated:** 5 sep 2003

**Journal:** Calc. Var. Partial Differential Equations

**Volume:** 17

**Pages:** 283 - 309

**Year:** 2003

**Abstract:**

It is proved that if $u\in BH^p(\Omega;R^d)$, with $p>1$, if $\{u_n\}$ is bounded in $BH^p(\Omega;R^d)$,
$

D^2_su_n

(\Omega)\to 0$, and if $u_n\rga u$ in $W^{1,1}(\Omega;R^d),$
then
$$ \int_{\Omega} f(x,u(x),\nabla u(x),\nabla^{2} u(x))\,dx \leq \liminf_{{n\to+\infty}
}
\int_{\Omega} f(x,u_{n}(x),\nabla u_{n}(x),\nabla^{2} u_{n}(x))\,dx$$
provided $f(x,u,\xi,\cdot)$ is 2-quasiconvex and satisfies some appropriate
growth and
continuity condition.
Characterizations of the 2-quasiconvex envelope when admissible test functions belong to $BH^p$ are provided.

**Keywords:**
Lower Semicontinuity, 2-quasiconvexity, functions with bounded Hessian, maximal function

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