0} \left\{ \begin{array}{rcl} -\Delta w_0 = \nabla_{w_0} F(w_0,w_0) +f_0& \text{in}&\Omega\\ \Delta w_0=\nabla_{w_0} F(w_0,w_0) +f_0& \text{in}&\Omega\\ \frac{\partial w_0}{\partial \nu}=\frac{\partial w_0}{\partial \nu}=0 & \text{on}