Calculus of variations, implicit partial differential equations and microstructure

We study existence of solutions for implicit partial differential equations of the form $\{ {\matrix{\displaystyle {F(x,u,Du)=0} \cr {{a}{.e}{.}\;{in}\;{\Omega }} \cr {u=\varphi } \cr {{on}\;\partial {\Omega }{.}} \cr} }.$ as well as minimization problems of the type $\displaystyle \inf \{ {{\textstyle\vint_{\Omega } {f(Du(x))dx:u=\varphi \;{on}\;\partial {\Omega }}} } \}\fleqno$ We discuss several examples that are relevant for applications to geometry, non linear elasticity or optimal design. All these examples exhibit what can be called microstructures. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)