Calculus of Variations and Partial Differential Equations

, Volume 1, Issue 2, pp 169-204

First online:

Singular perturbations as a selection criterion for periodic minimizing sequences

  • Stefan MüllerAffiliated withInstitut für Angewandte Mathematik, Universität Bonn

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Minimizers of functionals like\(\int_0^1 { \in ^2 u^2 _{xx} } + (u_x^2 - 1)^2 + u^2 dx\) subject to periodic (or Dirichlet) boundary conditions are investigated. While for ε=0 the infimum is not attained it is shown that for sufficiently small ε > 0, all minimizers are periodic with period ∼ ε1/3. Connections with solid-solid phase transformations are indicated.

Mathematics subject classification

34E15 49J45 (34C25, 35M10)