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Relaxation of convex functionals: the gap problem. (English) Zbl 1025.49012
The paper paper deals with integral representation properties of relaxed functionals of integral energies, and with the corresponding Lavrentiev phenomenon.
Let \(\Omega\) be a bounded domain in \({\mathbb R}^N\), let \(f\colon\Omega\times{\mathbb R}^{d\times N}\to[0,+\infty[\) be Carathéodory, satisfy the following growth conditions \[ z ^\alpha\leq f(x,z)\leq C(1+ z ^\beta)\text{ for all }z\in{\mathbb R}^{d\times N},\;{\mathcal L}^N\hbox{-a.e. }x\in\Omega \] for some \(C>0\) and \(1<\alpha\leq\beta<{N\alpha\over N-1}\), and let \(f\) be convex in the \(z\) variable for \({\mathcal L}^N\)-a.e. \(x\in\Omega\). For every open subset \(A\) of \(\Omega\) and \(u\in L^1(A;{\mathbb R}^d)\) let \[ {\mathcal F}(u,A)=\inf\left\{\liminf_{n\to+\infty}\int_A f(x,\nabla u(x))dx : \{u_n\}\subseteq W^{1,\beta}_{\text{loc}}(A;{\mathbb R}^d),\;u_n\to u\text{ in }L^1(A;{\mathbb R}^d)\right\}. \] In the paper it is first proved that if \(A\) is an open subset of \(\Omega\), and if \(u\in L^1(A;{\mathbb R}^d)\) satisfies \({\mathcal F}(u,A)<+\infty\), then \(u\in W^{1,\alpha}(A;{\mathbb R}^d)\) and for every open subset \(B\) of \(A\) \[ {\mathcal F}(u,A)=\int_B f(x,\nabla u(x))dx+\mu_s(u,B), \] where \(\mu_s(u,\cdot)\) is a nonnegative Radon measure singular with respect to \({\mathcal L}^N\).
Then, some conditions ensuring that \(\mu_s(u,\cdot)=0\), i.e., that no Lavrentiev phenomenon occurs, are proposed.
Finally, an example is discussed showing that, if \(f(x,\cdot)\) is no more convex but only quasiconvex for \({\mathcal L}^N\)-a.e. \(x\in\Omega\) and \(\alpha<\beta\), then the above representation result can be no more true.

49J45 Methods involving semicontinuity and convergence; relaxation
74B20 Nonlinear elasticity
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