Energy focusing in thin elastic structures and isometric immersions

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The present project's aim is to advance the mathematical understanding of thin elastic structures. It consists of two parts.In the first part, we consider variational problems that model sheets in the post-buckling regime. This regime is characterized by the focusing of (free) elastic energy in ridges and vertices. A better understanding of these phenomena is relevant not only in Applied Mathematics but also in Physics and Engineering.In our situation it is to be expected that the configurations observed in nature are close to minimizers of the free elastic energy. Hence we will investigate the minimization of elastic energy in various settings and derive scaling laws for the minimum of the energy in terms of the thickness of the sheet, which will be a small scalar parameter in the considered models. Our goal is to obtain a rigorous explanation for the emergence of post-buckled structures in this way.In the considered models, the leading order contribution to the elastic energy measures the deviation of the elastic deformation from an isometric immersion. (An immersion is isometric if the pull-back of the metric in the range coincides with the reference metric in the domain.) In this way, a natural connection to questions about isometric immersions appears, in particular about their uniqueness. The latter is the focus of the second part of the proposed project.The uniqueness of isometric immersions depends heavily on the required regularity. It has been conjectured that there is a critical regularity for isometric immersions above which isometric immersions are unique (up to rigid motions, and under suitable further hypotheses). It is known that the finiteness of a suitable notion of extrinsic curvature implies such uniqueness. Hence we will study the extrinsic curvature of immersions with low regularity. In coordinate charts, this amounts to the question whether or not certain distributional Jacobian and Hessian determinants belong to suitable function spaces. These distributional determinants are the central objects of interest in the second part of the proposal.Our maximal goal is to improve the known uniqueness results for isometric immersions with Hölder continuous derivatives by extending their range of validity to functions with lower regularity. The connection to the first part is to be found in the study of extrinsic curvature, which also plays an important role in the derivation of scaling laws.