Analysis of microstructure of a non-convex functional with penalization term

We study the properties of the minimizers of a singularly perturbed non-convex functional with a penalization term F ε , μ ( v , u ; I ) = 1 2 ε 2 ∫ I v x x 2 d x + 1 2 ∫ I W ( v x ) d x + μ 2 ε 2 ∫ I ( v − u ) 2 d x , where I = ( 0 , 1 ) , 0 < ε ≪ 1 and μ > 0 are parameters, u is a given function and W ( p ) = ( p 2 − 1 ) 2 . We analyze the phenomenon of microstructure formation and show the dependence on μ and u.