We consider the question raised in [1] of whether relaxed energydensities involving both bulk and surface energiescan be written as a sum of two functions, one depending on the net gradientof admissible functions, and the other on netsingular part.We show that, in general, they cannot. In particular, if the bulk densityis quasiconvex but not convex, thereexists a convex and homogeneous of degree 1 function of the jump such thatthere is no such representation.

Integral representation of relaxed energies and ofΓ-limits of functionals $$(u,v)\mapsto\int_\Omega f( x,u(x),v(x))\,dx$$ are obtained when sequences of fields v may develop oscillations and areconstrained to satisfya system of first order linear partial differential equations. Thisframework includes thetreatement of divergence-free fields, Maxwell's equations inmicromagnetics, and curl-freefields. In the latter case classical relaxation theorems in W1,p , arerecovered.

We give necessary and sufficient conditionswhich characterize the Young measuresassociated totwo oscillating sequences of functions, u n on $\omega_1\times \omega_2$ and v n on $\omega_2$ satisfying the constraint $v_n(y)=\frac{1}{|\omega_1|} \int_{\omega_1} u_n (x, y) dx$ . Our study is motivated by nonlinear effects induced by homogenization. Techniques based on equimeasurability and rearrangements are employed.

The problem of boundary stabilization for the isotropic linearelastodynamic system and the wave equation with Ventcel'sconditions are considered (see [12]). The boundaryobservability and the exact controllability were etablished in [11]. We prove here the enegy decay to zero for the elastodynamicsystem with stationary Ventcel's conditions by introducing anonlinear boundary feedback. We also give a boundary feedbackleading to arbitrarily large energy decay rates for theelastodynamic system with evolutive Ventcel's conditions. Aspectral study proves, finally, that the natural feedbackis not sufficient to assure the exponential decay in the case ofthe wave equation with Ventcel's conditions.