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A relaxation theorem in the space of functions of bounded deformation. (English) Zbl 0960.49014

The paper deals with the relaxation of the bulk energy \(\int_{\Omega} f({\mathcal E}u(x))dx\) in the space \(\text{BD}(\Omega)\) of the functions in \((L^1(\Omega))^N\) whose symmetric part of the distributional derivative \(Du\), \(Eu={Du+Du^T\over 2}\), is an \(N\times N\)-matrix valued Radon measure, \({\mathcal E}u\) is the absolutely continuous part with respect to Lebesgue measure of the deformation tensor \(Eu\), where \(\Omega\) is a bounded open subset of \(R^N\), and \(f\) is continuous and satisfies the following growth and coerciveness condition \[ {1\over C}|z|\leq f(z)\leq C(1+|z|)\text{ for every symmetric }N\times N\text{ matrix }z. \]
More precisely, given a regular bounded open set \(\Omega\), an integral representation result for the functional \[ \inf\left\{\liminf_{n\to\infty}\int_V f({\mathcal E}u_n(x))dx : u_n\in (W^{1,1}(V))^N,\;u_n\to u\text{ in }(L^1(V))^N\right\} \] defined on the set of all open subsets \(V\) of \(\Omega\), and \(u\in \text{SBD}(\Omega)\) is proved.
The integral representation result for the above functional turns out to involve both bulk and surface energy terms, together with the symmetric quasiconvex envelope of \(f\).
The main tools used to get the proof are a blow-up method, together with a Poincaré type inequality involving the operator \(E\) in place of the gradient.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
35J50 Variational methods for elliptic systems
49Q20 Variational problems in a geometric measure-theoretic setting
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References:

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