A unifying theme in this project is the asymptotic behavior of given equations as the domain approaches a "degenerate" region in the limit. The considered problems are nonlinear variational or partial differential equations, and the degeneracy in question can take various forms: loss of dimension, loss of regularity, or unboundedness. The first set of questions relates to the mathematical theory of nonlinear elasticity, which studies large mechanical deformations of three-dimensional elastic bodies. The investigator undertakes a long-term research program, with the scope of: deriving lower-dimensional shell theories through the methods of Gamma-convergence and understanding their connections with the geometry of the mid-surface, analyzing the (infinitesimal) isometries of surfaces and the effects of rigidity on the derived theories, studying nonlinear phenomena such as buckling and blistering for a given shell under compression (one application is related to plant growth). The second set of questions in this project relates to fluid dynamics. Boundary irregularities of various structures and scales are considered in the limit when the boundary behavior becomes degenerate. Other problems concern traveling fronts in combustion in unbounded channels, and the dynamics of solutions with large initial data under the Navier boundary conditions in thin three-dimensional shells.
Because elastic thin (or otherwise "degenerate") objects of various geometries are ubiquitous in the physical world, the precise understanding of laws governing their equilibria has many potential applications. For example, many growing tissues (leaves, flowers, or marine invertebrates) exhibit complicated configurations during their free growth and one would like to reproduce them with man-made means. A related long-standing problem in the mathematical theory of elasticity is to rigorously predict theories of such lower-dimensional objects starting from the nonlinear theory of full three-dimensional objects. For plates, a very recent effort has lead to rigorous justification of a hierarchy of such theories, depending on the magnitude of the applied forces and resulting in stretching, crumpling, bending, or a combination of these. For shells (when the mid-surface is curved), despite extensive use of their ad hoc generalizations in the literature and engineering applications, much less is known from the mathematical point of view. The investigator identifies several nonlinear problems in continuum mechanics of solids and fluid dynamics, naturally posed in specific degenerate domains, with the intention of rigorously understanding the behavior of the system based on general principles. At the heart of the program are interesting connections between calculus of variations, differential equations, geometry, material science, fluid dynamics, numerical analysis, and even biology. They have a potential to deliver useful observations in e.g. structural mechanics, while integrating the goal of exposing scientific results to a broader community at various education levels.
