The remarkable range of biological forms in and around us, such as the undulating shape of a leaf or flower in the garden, the coils in our gut, or the folds in our brain, raise a number of questions at the interface of biology, physics and mathematics. How might these shapes be predicted, and how can they eventually be designed? In this project, we furthered the current understanding of this problem, that brings together analysis, geometry and mechanics in the description of the morphogenesis of low-dimensional objects. Starting from the view that shape is the consequence of metric frustration in an ambient space, we examined the links between the classical Nash embedding problem and biological morphogenesis. Then, motivated by a range of experimental observations and numerical computations, we proved several rigorous results on curvature-driven patterning of thin elastic films, especially the asymptotic behaviors of the solutions as the (scaled) thickness becomes vanishingly small and the local curvature can become large. 

In addition to the above overarching theme of the project, we proved results in the following related contexts:

* Flexibility of the weak solutions to the Monge-Ampere system (MA) via convex integration. We formulated this new system of partial differential equations as an extension of the Monge-Ampere equation in d=2 dimensions, and showed how it naturally arises from the prescribed curvature problem, exhibiting its close relation to the classical problem of isometric immersions (II). Our main results achieved density, in the set of subsolutions, of the Holder regular C^{1,\alpha} solutions to the weak formulation (VK) of (MA), for all specific regularity regimes, in function of an arbitrary dimension d and codimension k of the problem. As an application of our results for (VK), we derived an energy scaling bound in the quantitative immersability of Riemannian metrics, for nonlinear energy functionals modeled on the energies of deformations of thin prestrained films in the nonlinear elasticity.

* Structure and properties of geodesics in kirigamized thin sheets. We studied how the presence of cuts in a thin planar sheet can alter its mechanical and geometrical response to loading. We used numerical experiments to characterize the geometric mechanics of kirigamized sheets as a function of the number, size and orientation of cuts. By varying the shape and number of the geodesics in a kirigamized sheet, we showed that we can control its deployment trajectory, and thence its functional properties as a robotic gripper or a soft light window. We proved that the family of all polygonal geodesics in a kirigamized sheet can be simultaneously rectified into a straight line by flat-folding the sheet so that its configuration is a piecewise affine planar isometric immersion

* Graph Laplacians on random data clouds. We studied Lipschitz regularity of elliptic partial differential equations on geometric graphs constructed from random data points, arising in data analysis in the context of graph-based learning, the main example of such being the equations satisfied by graph Laplacian eigenvectors. We proved high probability interior and global Lipschitz estimates for solutions of graph Poisson equations, relying on a probabilistic coupling argument of suitable random walks at the continuum level, and on an interpolation method for extending functions on random point clouds to the continuum manifold. As a byproduct of our general regularity results, we obtained high probability convergence rates for the convergence of graph Laplacian eigenvectors towards eigenfunctions of the corresponding weighted Laplace-Beltrami operators.

* Non-local Tug-of-War game with noise for fractional p-Laplace's operator. For the fractional p-s-Laplace operator, we introduced two families of non-local, non-linear averaging principles, in which the said operator emerges as the fractional order coefficient in the expansion of the deviation of each average from the value at the averaging center, and in the limit of the domain of averaging exhausting an appropriate cone. We established the dynamic programming principles modeled on the introduces averages, and showed that their solutions converge uniformly to viscosity solutions of the homogeneous non-local Dirichlet problem for the fractional p-s-Laplacian.

Last Modified: 04/05/2025
Modified by: Marta Lewicka