Equilibrium Validation for Triblock Copolymers via Inverse Norm Bounds for Fourth-Order Elliptic Operators

TL;DR

This paper introduces computer-assisted methods to rigorously analyze equilibrium solutions of triblock copolymers modeled by fourth-order PDEs, enabling validation of complex material behaviors.

Contribution

It develops a novel technique for bounding inverses of fourth-order elliptic operators, facilitating rigorous analysis of triblock copolymer equilibria.

Findings

Established a computer-assisted proof method for inverse norm bounds.

Applied the method to validate equilibrium states in triblock copolymers.

Demonstrated potential for verifying bifurcation points and continuation in related problems.

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, we demonstrate that 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.