Homogenized coefficients, quasiregular mappings and higher integrability of the gradients in two dimensional conductivity (Q2564717)

Let \(C=[0,1]^2\subset {\mathbb{R}}^2\), \(R_a(x)\), \(R_b(x)\) be \(C\)-periodic matrices which belong to \(SO(2)\) for all \(x\in {\mathbb{R}}^2\). Let \(C_1\), \(C_2\) be measurable disjoint sets in \(C\), \(C_1\cup C_2=C\), and \(\text{mes } C_1=f\in[0,1]\). Consider the following elliptic problem \((\ast)\) \(\sum_{i,j=1}^2 D_i(\sigma_{ij}(x)D_ju(x))=0\) where \[ \sigma(x)=\chi_{C_1}(x)\left ( \begin{matrix} \lambda_1(x)&0\\ 0&\lambda_2(x)\end{matrix} \right) + \chi_{C_2}(x)\left ( \begin{matrix} \eta_1(x)&0\\ 0&\eta_2(x)\end{matrix} \right), \] the constants \(\lambda_i\), \(\eta_i\) are such that \(\lambda_1\lambda_2>\eta_1\eta_2>0\), \(\lambda_2\geq\lambda_1\), \(\eta_2\geq\eta_1\). Let \(u_\xi\) be a solution of \((\ast)\) with \(C\)-periodic \(Du_{\xi}\) satisfying \(\int_C Du_{\xi}(y)dy=\xi\). The homogenized tensor \(H\) is defined as \(\sum_{h,k}H_{hk}\xi_k\xi_h=\int_C \sum_{ij} \sigma_{ij}(y)D_ju_{\xi}(y)D_iu_{\xi}(y) dy\). Let the eigenvalues \(h_1\) and \(h_2\) of \(H\) coinside: \(h_1=h_2=h\) and \(h_G(f)=h_G(\lambda_1,\lambda_2,\eta_1,\eta_2,f)= \sup_{C_1, \text{mes}(C_1)=f}h\). The author proves that if \(h_G(f)-h_G(0)= h_G(f)-\sqrt{\eta_1\eta_2}\sim a\cdot f^{\alpha}\) \((f\to 0)\) then \(\alpha=\sqrt{\eta_1/\eta_2}\).