Given the functional $J_A(y,u) = \frac{1}{2}||y-y_d||^2_{L^2(w,\omega)} + \frac{\lambda}{2}||u||^2_{L^2(w^{-1}, \omega)}$ where w is a function belongs to Muckenhoupt class. Given the optimization problem: min $J_A$ subject to-$div(A\triangledown{y}) = u$ in $\omega$. Assume matrix $A$ satisfies these inequalities for almost every $x\in \omega$, $w(x)|\zeta|^2 \leq C(\zeta^{T}\cdot A)$, $\ \zeta^{T}\cdot A \leq D(w(x)|\zeta|^2)$ for some constants $C$ and $D$$y = 0$ on $\partial \omega$, and the control constraints $u\in U_A$ where $U_A$ is a nonempty, closed and convex set of $L^{2}(w^{-1}, \omega)$. I'm interested in finding the dual problem + Lagrangian condition for this optimization problem. I also tried quite hard to prove the following inequality but still without any success: $(DJ_A(\bar{u}), (u-\bar{u}))\geq 0$ where $\bar{u} = $ optimal control solution to the optimization problem above.Hope someone can help me with either questions above. It's quite interesting, but also challenging, so I would really appreciate any help you can give.
Duality/Lagrangian condition and Variational Inequality of a cost functional
finite element;pde