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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
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References:

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