A variational approach to multiple layers of the bistable equation in long tubes
The author considers the elliptic boundary value problem \[ \triangle u-f(u)=0\;\text{in} \Omega,\;\partial u/\partial \nu =0\;\text{on} \partial\Omega, \tag{1} \] where \(f\) is a balanced bistable function and \(\Omega\) is an unbounded tube-shaped domain in \(\mathbb{R}^d\). Is is assumed that this domain is periodic along its axis, but perfect tubes, that is, \(\Omega=\mathbb{R}\times \Omega'\) for some \(\Omega'\subset \mathbb{R}^{d-1}\), are excluded. The associated energy functional is \((2)\;E(u)=\int_\Omega \;\left[{1\over 2}|\nabla u|^2+ W(u)\right] dx\), where \(W\) has a behaviour like \((u-1)^2(u+1)^2\). This problem is often used to study phase transitions. Since \(\Omega\) is not bounded, a Palais-Smale sequence, that is a bounded sequence \(u_n\) with \(E'(u_n)\to 0\), does in general not contain a convergent subsequence. Nevertheless, the author can prove by a series of interesting reasonings the existence of global minima of (2) in some subfunctionspaces. Under the assumption that these minima are isolated, the author constructs, by using these minima, an approximate solution. This function has the property that if there were no exact solution near this approximate solution then the negative gradient flow of the energy would significantly lower the energy, which leads to a contradiction.
scientific article; zbMATH DE number 1053361