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 $$ \int_\Omega f(x,u(x),\nabla u(x),\nabla² u(x))\,dx \leq \liminf_{{n\to+\infty}} \int_\Omega f(x,u_n(x),\nabla u_n(x),\nabla² u_n(x))\,dx$$ 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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