Analysis

The energy of the system is given by

$\mathcal{F} (u,\phi ) := \frac{1}{2} \kappa \int_\Omega |\Delta u |^2 +\frac{1}{2} \sigma \int_\Omega |\nabla u |^2 +\frac{1}{2} a \int_\Omega \phi^2 +\frac{1}{2} b \int_\Omega |\nabla \phi |^2 + c \int_\Omega \phi \Delta u$

and we will supplement the energy minimisation problem with two kinds of boundary conditions:

$\left( \left. u \right|_{\partial \Omega } , \, \left. \frac{\partial u}{\partial \nu} \right|_{\partial \Omega } \, \text{ and } \, \left. \phi \right|_{\partial \Omega } \right)$,
2. Navier boundary conditions

$\left( \left. u \right|_{\partial \Omega } , \, \left. \Delta u \right|_{\partial \Omega } \, \text{ and } \, \left. \phi \right|_{\partial \Omega } \right)$.