Variational methods for the analysis of geometric problems in solid mechanics

In this thesis I study geometric problems which commonly arise in material science. The goal of these models is to derive structural properties of the material by describing the time evolution of defects inside the material, and the overall shape evolution of the surface. My first result is a quantitative convergence result of solutions of the nonlocal Allen–Cahn equation to volume preserving mean curvature flow. Secondly, I prove a weak-strong uniqueness principle for the surface diffusion flow. The proof of these results is based on the relative entropy method, which uses gradient-flow calibrations and a Gronwall inequality. In my third result I study a problem in non-Euclidean elasticity for thin geodesic rods with thickness h. We prove a Gamma-convergence result of the elastic energy, rescaled by h to the 4-th power, as h tends to zero. Using rigidity estimates, I show that the deformations converge, to leading order, to a geodesic in the target space, and the energy converges to a functional measuring the difference between curvature tensors along the corresponding geodesic. The proof is based on uniformly Lipschitz approximations of the deformations, which guarantees that there exist suitable coordinate charts to work in a Euclidean setting.