Use the following potential energt expression of plate-bending and derive its governing differential equation and all possible boundary conditions and interpret them from the physics of the problem. 15 extra points for solving any plate problem with some boundary conditions and loading and comparing with an analytical solution.
Solve the calculus of variations that you have posed by writing down the first variation and Euler-Lagrange equations; obtain the differential equation, and the boundary conditions. Solve it analytically (if possible) or numerically.