The thin elastic membrane problem is another similar problem. See
Figure 1 for the geometry of the membrane.

The membrane is thin and elastic. It is initially planar and occupies
the 2D domain $\Omega$. It is fixed along part of its boundary $\Gamma _{u}$.
A transverse force $\mathbf {f}$ per unit area is applied. The final shape at
equilibrium is nonplanar. The final displacement of a point $\mathbf {x}$ on the
membrane is $\mathbf {u} (\mathbf {x} )$. There is no dependence on time.

The goal is to find the displacement $\mathbf {u} (\mathbf {x} )$ at equilibrium.

Figure 1. The membrane problem.

It turns out that the equations for this problem are the same as those
for the heat conduction problem - with the following changes:

The time derivatives vanish.

The balance of energy is replaced by the balance of forces.

The constitutive equation is replaced by a relation that states that the vertical force depends on the displacement gradient (${\boldsymbol {\nabla }}\mathbf {u}$).

If the membrane if inhomogeneous, the boundary value problem is: