Advertisement

Relaxation

  • Filip RindlerEmail author
Chapter
  • 3.2k Downloads
Part of the Universitext book series (UTX)

Abstract

Consider the functional
$$ \mathscr {F}[u] := \int _0^1 |u(x)|^2 + \bigl (|u'(x)|^2-1 \bigr )^2 \;\mathrm{d}x, \qquad u \in \mathrm {W}^{1,4}_0(0,1). $$
The gradient part of the integrand, \(a \mapsto (a^2-1)^2\), see Figure 7.1, has two distinct minima, which makes it a double-well potential . Approximate minimizers of \(\mathscr {F}\) try to satisfy \(u' \in \{-1, 1\}\) as closely as possible, while at the same time staying close to zero because of the first term.

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK

Personalised recommendations