Inverse norm bounds for fourth-order elliptic operators and equilibrium validation for triblock copolymers

73. Peter Rizzi, Evelyn Sander, Thomas Wanner:
    Inverse norm bounds for fourth-order elliptic operators and equilibrium validation for triblock copolymers
    Communications in Nonlinear Science and Numerical Simulation 115, Paper No. 106789, 27 pages, 2022.

Abstract

Block copolymers play an important role in materials sciences and have found widespread use in many applications. From a mathematical perspective, they are governed by a nonlinear fourth-order partial differential equation which is a suitable gradient of the Ohta-Kawasaki energy. While the equilibrium states associated with this equation are of central importance for the description of the dynamics of block copolymers, their mathematical study remains challenging. In the current paper, we develop computer-assisted proof methods which can be used to study equilibrium solutions in block copolymers consisting of more than two monomer chains, with a focus on triblock copolymers. This is achieved by establishing a computer-assisted proof technique for bounding the norm of the inverses of certain fourth-order elliptic operators, in combination with an application of a constructive version of the implicit function theorem. While these results are only applied to the triblock copolymer case, the obtained norm estimates can also be directly used in other contexts such as the rigorous verification of bifurcation points, or pseudo-arclength continuation in fourth-order parabolic problems.

Links

The preprint version of the paper can be downloaded from https://arxiv.org/abs/2203.10340, while the published version of the paper can be found at https://doi.org/10.1016/j.cnsns.2022.106789.

Bibtex

@article{rizzi:etal:22a,
   author = {Peter Rizzi and Evelyn Sander and Thomas Wanner},
   title = {Inverse norm bounds for fourth-order elliptic operators and
            equilibrium validation for triblock copolymers},
   journal = {Communications in Nonlinear Science and Numerical Simulation},
   year = 2022,
   volume = {115},
   pages = {Paper {N}o. 106789, 27 pages},
   doi = {10.1016/j.cnsns.2022.106789}
   }