\(G\)-convergence for non-divergence second order elliptic operators in the plane. (Q2249884)

A counterpart of the div-curl lemma for elliptic operators in non-divergence form is established in [\textit{V. V. Zhikov} and \textit{M. M. Sirazhudinov}, Math. USSR, Izv. 19, 27--40 (1982; Zbl 0494.35033)] in an \(L^2\)-setting. A generalization to an \(L^p\)-setting, \(p\neq 2\), in the plane is provided in the paper under review. More precisely, let \(\Omega \) be a bounded open subset of \(\mathbb {R}^2\) and \(K\geq 1\), and consider symmetric matrices \((A_k)_{k\geq 1}\) and \(A\) satisfying \[ | \xi | ^2/\sqrt {K} \leq \langle A_k(z)\xi, \xi \rangle,\quad \langle A(z)\xi, \xi \rangle \leq \sqrt {K}\;| \xi | ^2 \;\text{ and }\; \text{det}(A_k(z)) = \text{det}(A(z))=1 \] for a.e. \(z\in \Omega \) and all \(\xi \in \mathbb {R}^2\) and \(k\geq 1\). Fix \(K\in (1,2)\) and \(p=2K/(2K-1)\). Given non-negative functions \((v_k)_{k\geq 1}\), \(v\) in \(L_{\text{loc}}^p(\Omega)\) and functions \((u_k)_{k\geq 1}\), \(u\) in \(W_{\text{loc}}^{2,p}(\Omega)\), assume that \(D^2(A_k(\cdot ) v_k)=0\) for all \(k\geq 1\) and that \(v_k A_k(\cdot ) \rightharpoonup vA(\cdot)\) in \(L_{\text{loc}}^p(\Omega ;\mathbb {R}^{2\times 2})\) and \(u_k \rightharpoonup u\) in \(W_{\text{loc}}^{2,p}(\Omega)\). Then \[ \text{Tr}\left (v_k A_k(\cdot ) D^2 u_k \right ) \to \text{Tr}\left (v A(\cdot ) D^2 u \right ) \;\;\text{ in }\;\; \mathcal {D}'(\Omega_0) \] for any open subset \(\Omega_0\) of \(\Omega \). This result extends to arbitrary values of \(p\in (1,2)\) if the coefficients of \(A_k\) and \(A\) are VMO functions and the weak convergence of \((v_k A_k(\cdot ))\) holds in \(L_{\text{loc}}^2(\Omega)\). Applications to \(G\)-convergence results are given.