Bifurcation of a capillary minimal surface in a weak gravitational field

In this paper we study a variational elliptic boundary-value problem on a convex region $\Omega \subset \mathbb R^2$ with Bond parameter $\lambda \in \mathbb R$ that arises in hydromechanics and is closely related to the Plateau problem. It describes the behaviour of an elastic surface separating two liquid or gaseous media as the gravitational field changes. In the absence of gravitational force we have $\lambda =0$ and the solution to the problem is a minimal surface. Here we study the behaviour of this surface (loss of stability, bifurcations) when gravity is introduced. The method of analysis is based on reducing the problem to an operator equation in Hölder or Sobolev spaces with a non-linear Fredholm operator of index 0 that depends on the parameter $\lambda$, and applying the Crandall–Rabinowitz theorem on simple bifurcation points, the Lyapunov–Schmidt method of reduction to finite dimensions, and the key function method due to Sapronov. We obtain both necessary and sufficient general conditions for bifurcation, and study in detail the situation when $\Omega$ is a circle or a square.