Suffcient conditions for existence of solutions of a lower dimensional variational problem

Abstract

Solutions of variational problems under ane boundary conditions can be viewed equivalently as solutions of an implicit differential equation coupled with another equation that relates to quasiaffinity on the quasiconvex envelope of the integrand. In this work we derive conditions on an integrand and on its quasiconvex envelope which are sufficient for existence of solutions of a related lower dimensional variational problem derived through a dimension reduction process.