Structure of stable solutions of a one-dimensional variational problem

Yip, Nung Kwan

ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 4, pp. 721-751.

We prove the periodicity of all $H^2$-local minimizers with low energy for a one-dimensional higher order variational problem. The results extend and complement an earlier work of Stefan Müller which concerns the structure of global minimizer. The energy functional studied in this work is motivated by the investigation of coherent solid phase transformations and the competition between the effects from regularization and formation of small scale structures. With a special choice of a bilinear double well potential function, we make use of explicit solution formulas to analyze the intricate interactions between the phase boundaries. Our analysis can provide insights for tackling the problem with general potential functions.

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