The potential energy functional for a membrane stretched over a simply connected region of the plane can be expressed as
where is the deflection of the membrane, is the prescribed transverse pressure distribution, and is the membrane stiffness.
- The governing differential equation (Euler equation) for on .
- The permissible boundary conditions at the boundary of .
The principle of minimum potential energy requires that the functional be stationary for the actual displacement field . Taking the first variation of , we get
Plugging into the expression for ,
Now, the Green-Riemann theorem states that
where is the arc length around .
The potential energy function is rendered stationary if . Since is arbitrary, the condition of stationarity is satisfied only if the governing differential equation for on is
The associated boundary conditions are