documentclass[10pt,twoside]article usepackageamsfonts,amsfonts,amsthm,amssymb,amsmath numberwithinequationsection setlengthtextwidth6.5in setlengthtextheight9in setlengthoddsidemargin0in setlengthevensidemargin0in setlengthtopmargin-0.5in par newedtheoremtheoremTheorem[section]newedtheoremlemma[theorem]Lemmanewedtheoremproposition[theorem]Propositionnewedtheoremcorollary[theorem]Corollarynewedtheoremremarks[theorem]Remarksnewedtheoremremark[theorem]Remarknewedtheoremdefinition[theorem]Definitionnewedcommandrrmathbb R newedcommandAImathbb A newedcommandNmathbb N newedenvironment@abssec[1]par if@twocolumn par csname section*endcsname#1par else par vspace.05infootnotesize par parindent .2in par upshapebfseries #1. ignorespaces par fi par if@twocolumnelsepar vspace.1infi par newedcommandkeywordsnameKey words: par newedcommandAMSnameAMS subject classification par newedenvironmentkeywordspar if@twocolumn par section*keywordsnamepar else par vspace.05infootnotesize par parindent .2in par upshapebfseries keywordsname. ignorespaces par fi end@abssec par newedenvironmentAMSpar if@twocolumn par section*AMSnamepar else par vspace.05infootnotesize par parindent .2in par upshapebfseries AMSname. ignorespaces par fi end@abssec par begindocument par beginabstract In part I, it is shown that for integrals of the type beginequation* I(u,v):=int_Omega f(x,u(x),v(x))dx, endequation* with open, bounded, and $f:Omega times rr^m times rr^d to [0,+infty)$ Carathéodory satisfying a growth condition $0 le f(x,u,v) le C(1+{\vert v\vert}^p)$, for some $p in (1,+infty)$, a sufficient condition for lower semicontinuity along sequences $u_n to u$ in measure, $v_n weak v$ in $L^p$, ${cal A}v_n to 0$ in $W^{-1,p}$ is the ${cal A}_x$-quasiconvexity of $f(x,u,.)$. Here ${cal A}$ is a variable coefficients operator of the form beginequation* cal A:= sum_i=1^N A^(i)(x)frac partial partial x_i, endequation* with $A^{(i)} in C^{infty}(Omega;{cal M}^{l times d}) cap W^{1,infty}$, $i=1,..N$, satisfying the condition beginequation* rm rankleft( sum_i=1^N A^(i)(x)omega_i right)=rm constquad textfor $x in Omega$ and $omega in rr^N setminus \{0\}$, endequation* and ${cal A}_x$ denotes the constant coefficients operator one obtains by freezing $x$. Under additional regularity conditions on $f$ it is proved that the condition above is also necessary. A characterization of the Young measures generated by bounded sequences $\{v_n\}$ in $L^p$ satisfying the condition ${cal A}v_n to 0$ in $W^{-1,p}$ is obtained. par In part II, an integral representation for the functional beginequation* aligned cal F(m,M):= rm inf left{ liminf_k to +infty int_Omega f(x,m_k(x),nabla m_k(x))dx + int_Omegacap S(m_k) |[m_k](x)|dcal H^N-1 :right.
left.m_k in SBV(Omega;rr^N),quad |m_k(x)|=1quadtexta.e. in $Omega$,right.
left. m_k to mquad textinquad L^1(Omega;rr^N),quad nabla m_k weak Mquad textinquad L^2(Omega; rr^N)right} endaligned endequation* is obtained. This problem is motivated by equilibrium issues in micromagnetics. par In part III, the effective behavior of second order strain energy densities is obtained using relaxation and $G$ convergence techniques. The Cosserat theory is recovered within a dimension reduction analysis for $3 D$ thin domains with varying profiles. Homogeneous and inhomogeneous $2D$ models with periodic profiles are treated. par endabstract par enddocument