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

Irene Fonseca
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213 fonseca@andrew.cmu.edu

Giovanni Leoni
Dipartimento di Scienze e Tecnologie Avanzate
Università del Piemonte Orientale
Alessandria, Italy 15100
leoni@unipmn.it

and

Roberto Paroni
Dipartimento di Ingegneria Civile
Università degli Studi di Udine
Udine, Italy 33100
roberto.paroni@dic.uniud.it

ABSTRACT: It is proved that if $ u\in BH^p(\Omega ;$$ \mbox{${\mathbb{R}}^d$}$$ )$, with $ p>1$, if $ \{u_n\}$ is bounded in $ BH^p(\Omega ;$$ \mbox{${\mathbb{R}}^d$}$$ )$, $ \vert D^2_su_n\vert(\Omega)\to 0$, and if $ u_n\rightarrow u$ in $ W^{1,1}(\Omega ;$$ \mbox{${\mathbb{R}}^d$}$$ ),$ then

$\displaystyle \int_\Omega f(x,u(x),\nabla u(x),\nabla^2 u(x))\,dx \leq \liminf_{n\rightarrow +\infty} \int_\Omega f(x,u_n(x),\nabla u_n(x),\nabla^2 u_n(x))\,dx$

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.

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