Variational Approach to Geometric Function Theory

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The project is motivated by recent advances in the variational approach to Geometric Function Theory (GFT) and Nonlinear Elasticity (NE). The proposed problems of geometric nature originated from the Riemann Mapping Theorem; conformal mappings being univalent solutions of the Cauchy-Riemann system. Moving to the second order variational equations and their homeomorphic solutions offers new challenges. The goal is to characterize energy-functionals whose minimizers exist and resemble conformal maps. In our extremal problems mappings are free on the boundary. In such traction free problems we are concerned with existence and global invertibility of energy-minimal solutions. One needs to combine and develop the ideas of analysis and topology. There is a growing literature on inner variational equations that are weaker than the classical Euler-Lagrange equation. Even in the basic case of the Dirichlet integral there are many new surprising phenomena.GFT is currently a field of enormous activity where the general framework of NE is extremely fruitful and significant. The proposed variational approach contributes to this interplay and leads to concerted efforts of pure and applied mathematicians to work together. The research topics, and results already in place, have the potential impact on the development of elastic deformations, material science, continuum mechanics, etc. A thin elastic film tends to assume the shape of a minimal surface. The issue is to identify the optimal shape of a tubular thin film under stretch. This relates to the proposed studies of harmonic maps. There appear to be other applications in modeling cellular structures, foam physics and tissues as well. The PI will continue to host diverse groups of visiting scholars for mutual research.