论文标题
三阶椭圆运算符的逆标准范围对三嵌段共聚物的平衡验证
Equilibrium Validation for Triblock Copolymers via Inverse Norm Bounds for Fourth-Order Elliptic Operators
论文作者
论文摘要
块共聚物在材料科学中起着重要作用，并在许多应用中发现了广泛使用。从数学的角度来看，它们受非线性四阶部分微分方程的控制，该方程是Ohta-Kawasaki能量的合适梯度。尽管与该方程相关的平衡状态对于描述块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。在当前论文中，我们开发了计算机辅助的证明方法，这些方法可用于研究由两个以上单体链组成的块共聚物中的平衡溶液，重点是三嵌段共聚物。这是通过建立计算机辅助的证明技术来实现的，以界定某些四阶椭圆运算符的倒置的规范，并结合使用隐式函数定理的建设性版本的应用。尽管这些结果仅应用于三嵌段共聚物案例，但我们证明所获得的规范估计值也可以直接用于其他情况下，例如对分叉点的严格验证，或在第四阶抛物线寄生虫问题中持续进行伪估计。
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.
