Membrane Models: Variational Analysis and Large Deviations Principle in Subcritical Dimensions

In-depth understanding of the bending behavior of a membrane is often considered essential, e.g. for explaining cellular, as well as certain material properties and can lead to advancements in the fields of biology and materials science. In this thesis, two classical approaches to modeling a membrane are explored - a deterministic and a stochastic one.

We first consider a variational model for a membrane that is strongly attracted to two walls. We model the energy of such a membrane as an integral functional with a double-well potential (modeling the attraction to the two walls) and a |∆|^2-term (modeling the bending energy). We characterize optimal membrane shapes by means of Gamma-convergence.

In the stochastic setting, a membrane is modeled via a Gaussian process with covariance function being the Green's function of the Bi-Laplace operator. We show how this approach is linked to the variational model considered above via Large Deviations Principles. The incorporation of various constraints is also discussed.