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.

Keywords: higher order functional, local minimizer

@article{COCV_2006__12_4_721_0, author = {Yip, Nung Kwan}, title = {Structure of stable solutions of a one-dimensional variational problem}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {721--751}, publisher = {EDP-Sciences}, volume = {12}, number = {4}, year = {2006}, doi = {10.1051/cocv:2006019}, mrnumber = {2266815}, zbl = {1117.49025}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2006019/} }

TY - JOUR AU - Yip, Nung Kwan TI - Structure of stable solutions of a one-dimensional variational problem JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2006 SP - 721 EP - 751 VL - 12 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2006019/ DO - 10.1051/cocv:2006019 LA - en ID - COCV_2006__12_4_721_0 ER -

%0 Journal Article %A Yip, Nung Kwan %T Structure of stable solutions of a one-dimensional variational problem %J ESAIM: Control, Optimisation and Calculus of Variations %D 2006 %P 721-751 %V 12 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2006019/ %R 10.1051/cocv:2006019 %G en %F COCV_2006__12_4_721_0

Yip, Nung Kwan. Structure of stable solutions of a one-dimensional variational problem. ESAIM: Control, Optimisation and Calculus of Variations, Volume 12 (2006) no. 4, pp. 721-751. doi : 10.1051/cocv:2006019. http://www.numdam.org/articles/10.1051/cocv:2006019/

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