The proposal continues a long term research program of the PI to advance variational techniques in Geometric Function Theory with applied disciplines in mind. It provides an in-depth analysis of the concept of energy-minimal deformations between Euclidean domains (predominantly, traction free problems in dimension two). The ambition is to gain insights into the many new phenomena encountered in the Calculus of Variations, Minimal Surfaces, Nonlinear Elasticity, Material Science, and so forth. Historical example would be the celebrated mathematical structure -Teichmüller spaces, where one seeks to minimize the supremum norm of the distortion function. Other examples could be cited. We shall minimize the integral mean distortion instead. The existence and uniqueness of the mappings slipping along the boundary (traction free) with smallest mean distortion takes the challenge to a whole new level. The project continues the efforts of the PI to bring graduate students and postdoctoral scholars to Geometric Analysis with a wide range of applications, and to help them to mature meaningful interaction with physicists and engineers already in their post-doc phase. This is also a key to motivate young gifted students at school or college to enter mathematics. The program provides such an environment through educational materials (book in progress), Summer/Winter mini-courses, junior-senior seminars. PI has a history of an engaging research climate and strong international interaction on these efforts. He anticipates that the presence of applied fields in the proposal will improve the job opportunities for the PhD students and his numerous co-advisees.
