Speaker
Description
Motivated by variational models for heterogeneous biological membranes such as those described by the Canham-Helfrich functional, we study closed planar elastic curves whose bending stiffness depends on an additional scalar density. The resulting energy can be seen as the one-dimensional analogue of 2D membrane models.
Previous work on the static problem include \cite{bjss2023}, where the energy minimization under fixed length and total mass constraints is performed, followed by a bifurcation analysis around the trivial circular state is carried out. In \cite{dALR2024}, the authors study a related time-dependent problem, i.e. the associated $L^2$-gradient flow. Local well-posedness, global existence, and full convergence of the flow to a stationary solution are established, using a version of the Łojasiewicz–Simon gradient inequality.
This talk focuses on the qualitative properties of the $L^2$-gradient flow solutions, like the (non)preservation of convexity, embeddedness, positivity of the density, and rotational or axial symmetry of the initial datum. In particular, we identify conditions on the model parameters and the bending stiffness under which the asymptotic limit can be explicitly characterized as a homogeneous elastica, i.e., an elastica with constant density. Some numerical experiments will also be shown.
Bibliography
@article{bjss2023,
title = {Bifurcation of elastic curves with modulated stiffness},
volume = {34},
issn = {0956-7925, 1469-4425},
url = {https://www.cambridge.org/core/product/identifier/S0956792521000371/type/journal_article},
doi = {10.1017/S0956792521000371},
abstract = {We investigate the equilibrium configurations of closed planar elastic curves of fixed length, whose stiffness, also known as the bending rigidity, depends on an additional density variable. The underlying variational model relies on the minimisation of a bending energy with respect to shape and density and can be considered as a one-dimensional analogue of the Canham–Helfrich model for heterogeneous biological membranes. We present a generalised Euler–Bernoulli elastica functional featuring a density-dependent stiffness coefficient. In order to treat the inherent nonconvexity of the problem, we introduce an additional length scale in the model by means of a density gradient term. We derive the system of Euler–Lagrange equations and study the bifurcation structure of solutions with respect to the model parameters. Both analytical and numerical results are presented.},
language = {en},
number = {1},
urldate = {2026-03-20},
journal = {European Journal of Applied Mathematics},
author = {Brazda, K. and Jankowiak, G. and Schmeiser, C. and Stefanelli, U.},
month = feb,
year = {2023},
pages = {28--54},
}
@article{dALR2024,
title = {A dynamic approach to heterogeneous elastic wires},
volume = {392},
issn = {00220396},
url = {https://linkinghub.elsevier.com/retrieve/pii/S0022039624000664},
doi = {10.1016/j.jde.2024.02.001},
language = {en},
urldate = {2026-03-20},
journal = {Journal of Differential Equations},
author = {Dall'Acqua, Anna and Langer, Leonie and Rupp, Fabian},
month = may,
year = {2024},
pages = {1--42},
}
