The potential energy functional for a membrane stretched over a simply connected region ${\mathcal {S}}$ of the $x_{1}-x_{2}$ plane can be expressed as

where $w(x_{1},x_{2})$ is the deflection of the membrane, $p(x_{1},x_{2})$ is the prescribed transverse pressure distribution, and $\eta$ is the membrane stiffness.

Find:

The governing differential equation (Euler equation) for $w(x_{1},x_{2})$ on ${\mathcal {S}}$.

The permissible boundary conditions at the boundary $\partial {\mathcal {S}}$ of ${\mathcal {S}}$.

The principle of minimum potential energy requires that the functional $\Pi$ be stationary for the actual displacement field $w(x_{1},x_{2})$. Taking the first variation of $\Pi$, we get

where $s$ is the arc length around $\partial {\mathcal {S}}$.

The potential energy function is rendered stationary if $\delta \Pi =0$. Since $\delta w$ is arbitrary, the condition of stationarity is satisfied only if the governing differential equation for $w(x_{1},x_{2})$ on ${\mathcal {S}}$ is