Calculus of Variations and Geometric Measure Theory

I. Fonseca - G. Leoni - R. Paroni

On lower semicontinuity in $BH^p$ and 2-quasiconvexification

created on 22 Dec 2001
modified on 05 Sep 2003


Published Paper

Inserted: 22 dec 2001
Last Updated: 5 sep 2003

Journal: Calc. Var. Partial Differential Equations
Volume: 17
Pages: 283 - 309
Year: 2003


It is proved that if $u\in BH^p(\Omega;R^d)$, with $p>1$, if $\{u_n\}$ is bounded in $BH^p(\Omega;R^d)$, $
(\Omega)\to 0$, and if $u_n\rga u$ in $W^{1,1}(\Omega;R^d),$ then $$ \int\Omega f(x,u(x),\nabla u(x),\nabla2 u(x))\,dx \leq \liminf{n\to+\infty} \int\Omega f(x,un(x),\nabla un(x),\nabla2 un(x))\,dx$$ 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