The Effect of a Singular Perturbation to a 1-d Non-Convex Variational Problem

Nonconvex variational problems are of importance in modeling problems of microstructures and elasticity. In this book, we consider a 1--d nonconvex problem and we prove existence of solutions of the corresponding non--elliptic Euler--Lagrange equation by considering the Euler--Lagrange equation of the singular perturbed variational problem and passing to the linebreak limit. Under general assumptions on the potential we prove existence of Young--measure solutions. More restrictive conditions on the potential yield classical solutions via a topological method. The singular perturbed problem, which is also of interest for physicists due to the higher gradient surface--energy, is discussed in big detail.