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

Augsburger Schriften zur Mathematik, Physik und Informatik , Bd. 10

Markus Lilli

ISBN 978-3-8325-0928-6
100 pages, year of publication: 2005
price: 40.50 €

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 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.