A minimax estimation by incomplete data in the right-hand sides of elliptic second-order partial differential equations with discontinuous coefficients. (Q2739945)

Let \(\Omega_{i},\;i=1,2\) be open domains in \(\mathbb R^{n}, (n\leq3)\), such that \(\Omega_1\cap\Omega_2=\emptyset\), \(\bar\Omega_1\cap\bar\Omega_2\neq \emptyset\); \(\Gamma_{i}\) is a piecewise smooth boundary of \(\Omega_{i},\;i=1,2\), \(\gamma=\Gamma_1\cap\Gamma_2\). Let the state of system \(\phi_{i}(x)\in H^1(\Omega_{i}),\;i=1,2\) be defined as the generalized solution of transmission problem NEWLINE\[NEWLINE-\sum\limits_{i,j=1}^{n}\partial/ \partial x_{i}(a_{ij}^{(k)}(x)\partial \phi_{k}(x)/\partial x_{j})+ a_0^{(k)}(x)\phi_{k}(x)=f_{k}(x), x\in \Omega_{k};NEWLINE\]NEWLINE NEWLINE\[NEWLINE \partial\phi_{k}/\partial\nu_{A_{k}}+\sigma_{k}\phi_{k}=\psi_{k}, x\in\Gamma_{k}\setminus\gamma, k=1,2;NEWLINE\]NEWLINE NEWLINE\[NEWLINER_1(x)\partial\phi_{1}/\partial\nu_{A_{1}}+ R_2(x)\partial\phi_{2}/\partial\nu_{A_{2}}=\phi_1-\phi_2, \partial\phi_{2}/\partial\nu_{A_{2}}- \partial\phi_{1}/\partial\nu_{A_{1}}=\omega(x), x\in\gamma.NEWLINE\]NEWLINE Here \(f_{k}\in L^2(\Omega_{k}),\psi_{k}\in L^2(\Gamma_{k}\setminus\gamma), \omega\in L^2(\gamma), k=1,2\) are some given function; \(R_{k}(x), k=1,2\) are some continuous non-negative on \(\bar\gamma\) functions; \(\partial/\partial\nu_{A_{k}}\) is a normal derivative to \(\gamma\). Let us suppose that observation of the system state has the form \(y_{k}^{(i)}(x)=\int_{\Omega_{i}}g_{k}^{(i)}(x,y) \phi_{i}(y)dy+\xi_{k}^{(i)}(x), x\in\Omega_{i}, k=1,\ldots,m_{i}, i=1,2\), \(g_{k}^{(i)}(x,y)\in L^2(\Omega_{i}\times\Omega_{i}), i=1,2\) are given functions; \(\xi_{k}^{(i)}(x), i=1,2\) are some random fields. Under conditions that functions \(f_{i}(x), \phi_{i}(x), \omega(x), i=1,2\) and second moments of random fields \(\xi_{k}^{(i)}(x)\) are unknown exactly the authors obtain the minimax estimates for some functional on \(f_{i}(x)\).