Differential inclusions in quasiconformal analysis

Syracuse University, Syracuse NY

Investigators

Abstract

The proposal focuses on weakly differentiable mappings with first-order weak derivatives. This class includes, but is not limited to, quasiconformal and quasiregular mappings. A differential inclusion restricts the essential range of the derivative to a certain set of matrices. Main sources of differential inclusions are differential geometry and the theory of nonlinear partial differential equations. Typically, various notions of convexity and connectedness in the matrix space drive the analysis of existence and regularity of solutions. One of basic questions is which differential inclusions imply local invertibility. Invertibility is often established in two steps: first it is shown that the mapping is discrete and open (that is, a branched cover); the second step is to prove that the branch set is empty. Implementation of each step encounters problems that continue to challenge the available methods of analysis and topology. The proposal also brings the tools of geometric function theory into the field of ordinary differential equations. In the presence of canonical coordinates on the Euclidean space the driving vector field in an autonomous system of differential equations can be identified with a mapping, and the geometry of this mapping turns out to be related to the uniqueness of solution. Thirdly, the techniques introduced in quasiconformal analysis are also effective in the studies of smooth (e.g., harmonic) mappings, which in turn find applications in the theory of minimal surfaces Minimal surfaces are mathematical models of thin films, for instance soap bubbles. Our understanding of their shapes develops through the solution of extremal problems. For example, how far apart can one move two boundary curves of a minimal surface before the surface breaks down? The principal investigator will apply the techniques of geometric function theory to such extremal problems. This approach is not limited to models of thin films and is also relevant in the studies of elastic deformation of solid materials. Another part of the proposal addresses uniqueness and stability of solutions of ordinary differential equations, which are commonplace in physics and engineering. They appear as equations of motion for one or several particles, with the number of particles affecting the dimensionality of the problem and the geometry of vector fields involved. Geometric function theory allows one to establish the uniqueness of a solution in situations where the standard results of the theory of ordinary differential equations do not apply. The PI works with post-docs and graduate students and organizes the Syracuse Analysis Study group.