The author proves an homogenization result for a von Karman plate, using \( \Gamma \)-convergence tools. He starts with the thin domain \(\Omega ^{\varepsilon }=\omega \times h(-\frac{1}{2},\frac{1}{2})\) with \(h>0\) and \( \omega \subset \mathbb{R}^{2}\). He first defines an admissible composite material of class \(\mathcal{W}(\eta _{1},\eta _{2},\rho )\), \(0<\eta _{1}\leq \eta _{2}\) and \(\rho >0\), as a family \((W^{h})_{h}\) of energies from \(\Omega \times \mathbb{R}^{3\times 3}\) to \([0,+\infty ]\) such that \(W^{h}\) is almost surely equal to a Borel function on \(\Omega \times \mathbb{R}^{3\times 3}\), \( W^{h}\) is frame-indifferent, non-degenerate, minimal at \(\mathrm{Id}\) and has a quadratic approximation \(Q^{h}\) that is \[ \sup_{h>0}\mathrm{ess}\sup_{x\in \Omega }|W^{h}(x,\mathrm{Id}+G)-Q^{h}(x,G)|\leq |G|^{2}r(|G|) \] for a monotone function \(r: \mathbb{R}^{+}\rightarrow [ 0,+\infty ]\) with \(r(\delta )\rightarrow 0\) as \(\delta \rightarrow 0\). The associated energy is defined through \( I^{h}(y)=\frac{1}{h^{4}}\int_{\Omega }W^{h}(x,\nabla _{h}y(x))dx\) for \(y\in H^{1}(\Omega ,\mathbb{R}^{3})\) with \(\nabla _{h}y=(\partial _{1}y,\partial _{2}y,\frac{1}{h}\partial _{3}y)\). The author then defines the notion of convergence for a sequence \((y^{h})_{h}\subset L^{2}(\Omega ,\mathbb{R}^{3})\) to a triple \((\overline{R},u,v)\in \mathrm{SO}(3)\times L^{2}(\omega ,\mathbb{R} ^{2})\times L^{2}(\omega )\)if there exist rotations \((\overline{R}^{h})_{h} \) and functions \((u^{h})_{h}\subset L^{2}(\omega ,\mathbb{R}^{2})\) and \( (v^{h})_{h}\subset L^{2}(\omega )\) such that \[ (\overline{R}^{h})^{T}(\int_{(- \frac{1}{2},\frac{1}{2})}y^{h}(x^{\prime },x_{3})dx_{3}-\int_{\Omega }y^{h}dx)=\begin{pmatrix} x^{\prime }+h^{2}u^{h}(x^{\prime }) \\ hv^{h}(x^{\prime })\end{pmatrix}, \] \(u^{h}\rightarrow u\) in \(L^{2}(\omega ,\mathbb{R}^{2})\), \( v^{h}\rightarrow v\) in \(L^{2}(\omega )\) and \(\overline{R}^{h}\rightarrow \overline{R}\).
The main result of the paper proves the existence of a subsequence \((h_{n})_{n}\) and of a limit energy functional \(I^{0}:\mathcal{A} (\omega )\rightarrow \mathbb{R}_{0}^{+}\), with \(\mathcal{A}(\omega )=H^{1}(\omega ,\mathbb{R}^{2})\times H^{2}(\omega )\), such that:
1) if \((y^{h_{n}})_{n}\subset H^{1}(\omega ,\mathbb{R}^{2})\) is such that \( \lim \sup_{n}I^{h_{n}}(y^{h_{n}})<+\infty \), there exists \((\overline{R} ,u,v)\in \mathrm{SO}(3)\times \mathcal{A}(\omega )\), and \((y^{h_{n}})_{n}\) converges to \((\overline{R},u,v)\) in the sense of the preceding definition;
2) if \((y^{h_{n}})_{n}\subset H^{1}(\omega ,\mathbb{R}^{2})\) satisfies \(\lim \sup_{n}I^{h_{n}}(y^{h_{n}})<+\infty \) and \((y^{h_{n}})_{n}\) converges to \(( \overline{R},u,v)\), then \(\lim \inf_{n}I^{h_{n}}(y^{h_{n}})\geq I^{0}(u,v)\);
3) for every \((\overline{R},u,v)\in SO(3)\times \mathcal{A}(\omega )\) there exists a sequence \((y^{h_{n}})_{n}\subset H^{1}(\omega ,\mathbb{R}^{2})\) such that \((y^{h_{n}})_{n}\) converges to \((\overline{R},u,v)\) and \( \lim_{n}I^{h_{n}}(y^{h_{n}})=I^{0}(u,v)\).
\(I^{0}\) is given through \(I^{0}(u,v)=\int_{\omega }Q(x^{\prime },\mathrm{sym}\nabla u+ \frac{1}{2}\nabla u\otimes \nabla v,-\nabla ^{2}v)dx^{\prime }\). For the proof, the author analyzes the properties of minimizing sequences within this context and he adapts the \(\Gamma \)-convergence techniques. He also uses decomposition results from G. Griso [Anal. Appl., Singap. 6, No. 1, 11–22 (2008; Zbl 1210.74121)], previous results he proved and properties of equiintegrable functions which are proved in an appendix.