Extremal Problems in Quasiconformal Geometry and Nonlinear PDEs, an Invitation to n- Harmonic Hyperelasticity

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In this project the principal investigator initiates a study of extremal problems for mappings of finite (usually unbounded) distortion, hoping to provide background for further developments in geometric function theory and nonlinear elliptic partial differential equations. The plan of attack relies on geometric and physical intuition, drawn especially from the theory of nonlinear elasticity and materials science. This represents a trend that in recent years has become more pronounced, and it has led to increasing efforts by pure and applied mathematicians to combine such ideas and results. The project centers on the remarkable relationship (first envisioned by the principal investigator and Jani Onninen) between mappings of finite distortion and hyperelasticity. Both theories are governed by variational principles and problems of compelling common mathematical interest. The new fronts that have been created in geometric function theory include polyconvex integrals of the distortion function and the associated total n-harmonic energy. Of special interest are mappings of smallest mean distortion, whose existence, regularity, and global invertibility are related to deep unsolved mathematical questions. For instance, the present proposal takes on difficult questions concerning deformations of spherical rings in Euclidean n-space, and a much studied conjecture of Nitsche (1962). This is a tentative first step toward an n-dimensional theory of moduli, a subject that hopefully will develop into a very coherent analogue of Teichmuller theory in higher dimensions. The project features new concepts (such as free Lagrangians) and challenging questions galore, some already prepared for answers, others of a more speculative, long-term character. With its elaborate design, this project will encourage the dissemination of modern geometric function theory, a classical field that has undergone a tremendous transformation in recent years, to a wider audience and will enhance the understanding of the subject's pervasive presence in applications. The project reflects continued efforts by the principal investigator to provide an active and welcoming research environment for graduate students and young scholars through effective training, the creation of educational materials, and the fostering of scientific partnerships both within the U.S. and overseas. This endeavor has hitherto been particularly effective for groups historically underrepresented in mathematics, especially women. The proposed problems will find considerable interest among geometric analysts, especially researchers in such physically relevant fields as partial differential equations, the calculus of variations, and nonlinear elasticity theory.