Bifurcation problems for differential geometric partial differential equations

• Applicant: Professor Dr. Hannes Uecker
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 563415565

Differential geometric partial differential equations (PDE) occur in both, pure mathematics and applications, in particular in material science and biology. The simplest examples are constant mean curvature (CMC) surfaces, e.g., soap bubbles. These surfaces arise inter alia as area minimizers under a volume constraint, and the associated PDEs can be classified as second order quasilinear elliptic. Classes of 4th order geometric PDEs for instance occur in the modeling of cell membranes. CMC surfaces and the associated PDEs are intensely studied since the 18th century, but compared to other classical (semilinear) PDEs the theory is less developed, due to the stronger type of nonlinearity and the geometric constraints. Specifically for 4th order equations only few basic short time existence results for the flow are known, and few basic steady solutions and their stability, usually with some axial symmetry (surfaces of revolution), and there exist only few studies of the parameter dependence and bifurcations of non axially symmetric solutions. The goal of the project is to combine analysis and numerics to obtain new stability and bifurcation results for biologically motivated differential geometric PDEs. On the numerical side we shall apply and further develop the tool pde2path, and the analysis will focus on amplitude equations and modulation equations.

• DFG Programme: Research Grants