# shape of membrane on circular frame with pressure difference

The equation describing membranes separating two fluids with a pressure differential is the Young-Laplace equation first considered in the early 1800s. In modern geometric language the equation states that the mean curvature of the membrane is proportional to the pressure differential.

In the case of the constant pressure differential and a circular boundary, just as the case of the meniscus in capillary surfaces, the solution is a portion of the round sphere. The radius of the sphere depends on the elasticity constant of the membrane and the pressure differential.

For general membranes (idealized to 2D sheet):

Suppose the configuration "at rest" is that of a unit disc $D = \{x\in \mathbb{R}^2: |x| \leq 1\}$. The deformation map is $$\Phi: D \to \mathbb{R}^3$$ with the condition $\Phi |_{\partial D}$ is the expected embedding of the unit circle into the $x_1-x_2$ plane in $\mathbb{R}^3$. The local elastic energy density is (under isotropic assumption) $$e[\Phi] = f(|\nabla \Phi|^2, |\partial_1 \Phi \wedge \partial_2 \Phi|^2)$$ where the function $f$ models the material properties of the membrane

The case where $f(a,b) = \sqrt{b}$ is the soap-film model (minimizing area under constraint). By changing the function $f$ you get different models of elastic membranes.

Once you have specified a function $f$, the situation you are interested in is described by thinking in terms of calculus of variations:

1. Let $E[\Phi] = \int_D e[\Phi]~\mathrm{d}A$ be the total energy. We take its $L^2$ gradient $\nabla_\Phi E$ by defining it as the function satisfying $$\int_D (\nabla_\Phi E)\psi ~\mathrm{d}A = \delta E[\Phi][\psi]$$ for every $\psi:D\to\mathbb{R}^3$ that vanishes on the boundary. Here $\delta E$ is the variation of $E$.
2. The pressure gives force per unit area; but the area is the stretched area. So the force density when translated into the parametrisation by the rest state is in fact $p(\Phi) |\partial_1\Phi \wedge \partial_2\Phi|$, the pressure times the area density.
3. Force balance requires then $$p | \partial_1\Phi \wedge \partial_2\Phi| = \nabla_\phi E$$ (basically the Euler-Lagrange equation with constraint); I removed the dependence of the pressure $p$ on position $\Phi$ since you've stated that we are facing a constant pressure differential. This equation is in general a nonlinear system of elliptic partial differential equations. The precise form of this equation depends, again, on the function $f$ that you choose.

In general probably this system of equations cannot be solved explicitly.