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.