Structure of stable solutions of a one-dimensional variational problem

Nung Kwan Yip
2006 E S A I M: Control, Optimisation and Calculus of Variations  
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
more » ... ce 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. Mathematics Subject Classification. 47J20, 49K20, 34K26. The reason for this choice is that the corresponding Euler-Lagrange equation for (1) is given by a linear differential equation with constant coefficients together with some linear jump conditions for the solutions. As our goal is to investigate the relation between the critical points of E and periodic patterns, we first present the solution of a unit cell problem. For each l > 0, let P (x, l) be the function which solves the following
doi:10.1051/cocv:2006019 fatcat:cteiiwktqngg3nihqpqzbdqcui