Generalized monotonicity from global minimization in fourth-order ordinary differential equations (Q2757157)

This paper deals with the solutions of the stationary extended Fisher-Kolmogorov equation with general potential that are global minimizers of an associated variational problem. The aim of the author is to demonstrate how the global minimization property imposes a form of monotonicity on the solution, thereby drastically limiting the set of potential global minimizers. To some extent this concept is modelled on the case of second-order problems in \(\mathbb{R}^n\), where symmetrization techniques can be applied to prove the global minimizers are radially symmetric and monotonic in the radial variable.NEWLINENEWLINENEWLINEThe author illustrates the concept on two model systems:NEWLINENEWLINENEWLINE(a) a model of an elastic strict supported by an elastic foundation;NEWLINENEWLINENEWLINE(b) a model of pattern formation in polymeric materials under tension.