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
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