Energy focusing in thin elastic structures and isometric immersions
Final Report Abstract
The project was devoted to the study of mathematical properties of thin elastic structures. Several aspects were considered. Firstly, we considered elastic structures in the so-called post-buckling regime. The analysis was performed under additional simplifying assumptions, such as the convexity of certain components of the elastic deformation. Secondly, we proved certain identities from geometric analysis in function spaces of weak regularity, which are closely connected to the theory of elasticity. Thirdly, we showed how certain discrete models of elastic plates approximate continuous models in the limit of vanishing lattice size. Finally, we showed that a certain model for inhomogeneous elastic plates involving a nonlinear geometric curvature term and an additional penalising term for the non-flat part, converges in a variational sense to a well-defined limit problem as the penalisation increases to infinity.
Publications
- Energy scaling for a conically constrained sector
Peter Gladbach and Heiner Olbermann
(See online at https://doi.org/10.48550/arXiv.1907.06109) - On a Γ-limit of willmore functionals with additional curvature penalization term. SIAM Journal on Mathematical Analysis, 51(3):2599–2632, 2019
Heiner Olbermann
(See online at https://doi.org/10.1137/18m1203596) - Coarea formulae and chain rules for the Jacobian determinant in fractional Sobolev spaces. Journal of Functional Analysis, 278(2):108312, 2020
Peter Gladbach and Heiner Olbermann
(See online at https://doi.org/10.1016/j.jfa.2019.108312) - Approximation of the Willmore energy by a discrete geometry model. Advances in Calculus of Variations, 2021
Peter Gladbach and Heiner Olbermann
(See online at https://doi.org/10.1515/acv-2020-0094)
