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Calculus of variations, implicit partial differential equations and microstructure
Author(s) -
Dacorogna Bernard
Publication year - 2006
Publication title -
gamm‐mitteilungen
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.239
H-Index - 18
eISSN - 1522-2608
pISSN - 0936-7195
DOI - 10.1002/gamm.201490028
Subject(s) - omega , partial derivative , partial differential equation , mathematics , elasticity (physics) , matrix (chemical analysis) , calculus (dental) , mathematical analysis , chemistry , physics , thermodynamics , medicine , dentistry , quantum mechanics , chromatography
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)