Local Minimizers of Singularly Perturbed Functionals with Nonlocal Term

Variational problems involving nonlocal terms are of growing importance in the mathematical modeling of microstructures that occur in solid-solid phase transitions. For a model describing an austenite / twinned martensite interface, it is conjectured that the minimizers are highly oscillating near-periodic functions. The nonlocality, however, turns this problem into a very difficult one.

In this book, a similar problem is considered. It is given by singularly perturbed functionals including a double-well potential and a quite general nonlocal term with bounded kernel. Using the concept of Gamma-convergence and the Modica-Mortola theorem, the existence of near-periodic local minimizers is shown.