Analysis of Multi-phase Vesicles and Membranes

Membranes and vesicles undoubtedly play a central role in biology, physics, and medicine, and a profound understanding of their mathematical foundations is a key goal of research. However, the complexity of the underlying equations poses a challenge and complicates their comprehensive understanding, particularly when considering membranes with additional structures, rather than focusing solely on homogeneous membranes. In recent decades, homogeneous cell membranes have been intensively studied within mathematics, leading to numerous breakthroughs such as the proof of the Willmore conjecture in 2012 by Fernando Marques and Andre Neves. Our understanding of the dynamics of cell membranes has greatly advanced through groundbreaking work by Mayer, Simonett, Escher, Kuwert, Schätzle, and numerous other scientists. Our goal is to take the logical next step from homogeneous membranes to membranes with additional structures. Specifically, we will investigate cell membranes composed of two or more different materials, with a particular focus on their temporal evolution. We model the behavior of the membranes using the Canham-Helfrich functional with varying parameters reflecting the different properties of the materials. Additionally, at the interfaces between the membranes, we consider the elastic energy or the length of these curves. The main objective of this research project is to obtain initial analytical results for the underlying highly complex equations. This includes examining the local existence of solutions, evolutionary stability of local minima, and analyzing possible singularities in finite or infinite time. The underlying analytical equations have been practically unexplored for obvious reasons: they are critical quasilinear equations on the various phases of the cell, coupled with additional quasilinear parabolic equations on the boundary describing its temporal evolution. Numerical experiments in recent years suggest that these equations are well-posed and exhibit interesting new qualitative properties. We aim to gain new insights into the complex dynamics of such systems and provide analytical results that have not been available previously. Our goal is not only to explore the boundaries of geometric analysis in this area but also to significantly expand them.