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.

嵌段共聚物在材料科学中发挥着重要作用，并已在许多应用中得到广泛应用。从数学的角度来看，它们由非线性四阶偏微分方程控制，该方程是 Ohta-Kawasaki 能量的合适梯度。虽然与该方程相关的平衡状态对于描述嵌段共聚物的动力学至关重要，但它们的数学研究仍然具有挑战性。在本文中，我们开发了计算机辅助证明方法，可用于研究由两个以上单体链组成的嵌段共聚物的平衡解，重点是三嵌段共聚物。这是通过建立一种计算机辅助证明技术来限制某些四阶椭圆算子的逆范数来实现的，结合隐函数定理的构造版本的应用。虽然这些结果仅适用于三嵌段共聚物的情况，但所获得的范数估计也可以直接用于其他情况，例如分叉点的严格验证，或四阶抛物线问题中的伪弧长延拓。