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Relaxation of multiple integrals below the growth exponent. (English) Zbl 0868.49011
Summary: The integral representation of the relaxed energies $\begin{split}{\mathcal F}^{q,p}(u,\Omega):=\inf_{\{u_n\}} \Biggl\{\liminf_{n\to\infty} \int_\Omega F(x,u_n,\nabla u_n) dx: u_n\in W^{1,q}(\Omega,\mathbb{R}^d),\\ u_n\rightharpoonup u\text{ weakly in }W^{1,p}(\Omega,\mathbb{R}^d)\Biggr\},\end{split}$
$\begin{split} {\mathcal F}^{q,p}_{\text{loc}}(u,\Omega):= \inf_{\{u_n\}} \Biggl\{\liminf_{n\to\infty} \int_\Omega F(x,u_n,\nabla u_n) dx: u_n\in W^{1,q}_{\text{loc}}(\Omega,\mathbb{R}^d),\\ u_n\rightharpoonup u\text{ weakly in }W^{1,p}(\Omega,\mathbb{R}^d)\Biggr\}\end{split}$ of a functional $E: u\mapsto \int_\Omega F(x,u,\nabla u) dx,\quad u\in W^{1,q}(\Omega,\mathbb{R}^d),$ where $$0\leq F(x,\zeta,\xi)\leq C(1+|\zeta|^r+ |\xi|^q)$$ and $$\max\{1,r{N-1\over N+r},q{N-1\over N}\}< p\leq q$$, is studied. In particular, $$W^{1,p}$$-sequential weak lower semicontinuity of $$E(\cdot)$$ is obtained in the case where $$F=F(\xi)$$ is a quasiconvex function and $$p>q(N-1)/N$$.

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation 49Q10 Optimization of shapes other than minimal surfaces
##### Keywords:
quasiconvexity; relaxation; weak lower semicontinuity
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##### References:
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