Nonlinear Partial Differential Equations: Singularities, Potential Theory, and Geometric Applications

The main objective of the project is to study properties of solutions to fully nonlinear, elliptic partial differential equations. Rather than studying more traditional existence-uniqueness problems the main task will be to investigate qualitative questions. These concern the behaviour of solutions to the equations, the description of possible pathologies and singularities the solutions can have, and conditions for the absence of singularities. Understanding of the singular behaviour of solutions is very important for applications in geometry, physics, elasticity, and mechanics. From this point of view, probably the most important problem is to find explicit information about singularities of solutions.