3D-2D asymptotic analysis for thin structures rests on the mastery of scaled gradients $\left({\nabla}_{\alpha}{u}_{\epsilon}|\frac{1}{\epsilon}{\nabla}_{3}{u}_{\epsilon}\right)$ bounded in ${L}^{p}(\Omega ;{\mathbb{R}}^{9}),\phantom{\rule{4pt}{0ex}}1\<p\<+\infty .$ Here it is shown that, up to a subsequence, ${u}_{\epsilon}$ may be decomposed as ${w}_{\epsilon}+{z}_{\epsilon},$ where ${z}_{\epsilon}$ carries all the concentration effects, i.e. $\left\{{\left|\left({\nabla}_{\alpha}{w}_{\epsilon}|\frac{1}{\epsilon}{\nabla}_{3}{w}_{\epsilon}\right)\right|}^{p}\right\}$ is equi-integrable, and ${w}_{\epsilon}$ captures the oscillatory behavior, i.e. ${z}_{\epsilon}\to 0$ in measure. In addition, if $\left\{{u}_{\epsilon}\right\}$ is a recovering sequence then ${z}_{\epsilon}={z}_{\epsilon}\left({x}_{\alpha}\right)$ nearby $\partial \Omega .$

3D-2D asymptotic analysis for thin structures rests on the mastery
of scaled gradients $\left({\nabla}_{\alpha}{u}_{\epsilon}|\frac{1}{\epsilon}{\nabla}_{3}{u}_{\epsilon}\right)$ bounded in ${L}^{p}(\Omega ;{\mathbb{R}}^{9}),\phantom{\rule{4pt}{0ex}}1<p<+\infty .$ Here it is shown that, up to a
subsequence, ${u}_{\epsilon}$ may be decomposed as ${w}_{\epsilon}+{z}_{\epsilon},$
where ${z}_{\epsilon}$ carries all the concentration effects,
$\left\{{\left|\left({\nabla}_{\alpha}{w}_{\epsilon}|\frac{1}{\epsilon}{\nabla}_{3}{w}_{\epsilon}\right)\right|}^{p}\right\}$ is
equi-integrable, and ${w}_{\epsilon}$ captures the oscillatory behavior,
${z}_{\epsilon}\to 0$ in measure. In addition, if $\left\{{u}_{\epsilon}\right\}$ is
a recovering sequence then ${z}_{\epsilon}={z}_{\epsilon}\left({x}_{\alpha}\right)$ nearby
$\partial \Omega .$

DiPerna's and Majda's generalization of Young measures
is used to describe oscillations and concentrations in sequences of maps ${\left\{{u}_{k}\right\}}_{k\in \mathbb{N}}\subset {L}^{p}(\Omega ;{\mathbb{R}}^{m})$ satisfying a linear differential constraint $\mathcal{A}{u}_{k}=0$. Applications to sequential weak lower semicontinuity of integral functionals on $\mathcal{A}$-free sequences and to weak continuity of determinants are given. In particular, we state necessary and sufficient conditions for weak* convergence of det$\nabla {\varphi}_{k}\stackrel{*}{\rightharpoonup}\mathrm{det}\nabla \varphi $ in measures on the closure of $\Omega \subset {\mathbb{R}}^{n}$ if ${\varphi}_{k}\rightharpoonup \varphi $ in ${W}^{1,n}(\Omega ;{\mathbb{R}}^{n})$. This convergence holds, for example, under...

Local Lipschitz continuity of minimizers of certain integrals of the Calculus of Variations is obtained when the integrands are convex with respect to the gradient variable, but are not necessarily uniformly convex. In turn, these regularity results entail existence of minimizers of variational problems with non-homogeneous integrands nonconvex with respect to the gradient variable. The $x$-dependence, explicitly appearing in the integrands, adds significant technical difficulties in the proof.

A characterization of the total variation $TV\left(u,\mathrm{\Omega}\right)$ of the Jacobian determinant $detDu$ is obtained for some classes of functions $u:\mathrm{\Omega}\to {\mathbb{R}}^{n}$ outside the traditional regularity space ${W}^{1,n}\left(\mathrm{\Omega};{\mathbb{R}}^{n}\right)$. In particular, explicit formulas are deduced for functions that are locally Lipschitz continuous away from a given one point singularity ${x}_{0}\in \mathrm{\Omega}$. Relations between $TV\left(u,\mathrm{\Omega}\right)$ and the distributional determinant $\text{Det}Du$ are established, and an integral representation is obtained for the relaxed energy of certain polyconvex functionals at maps $u\in {W}^{1,p}\left(\mathrm{\Omega};{\mathbb{R}}^{n}\right)\cap {W}^{1,\mathrm{\infty}}\left(\mathrm{\Omega}\left\{{x}_{0}\right\};{\mathbb{R}}^{n}\right)$.

Integral representation of relaxed energies and of
-limits of functionals
$$(u,v)\mapsto {\int}_{\Omega}f(x,u\left(x\right),v\left(x\right))\phantom{\rule{0.166667em}{0ex}}dx$$
are obtained when sequences of fields may develop oscillations and are
constrained to satisfy
a system of first order linear partial differential equations. This
framework includes the
treatement of divergence-free fields, Maxwell's equations in
micromagnetics, and curl-free
fields. In the latter case classical relaxation theorems in , are
recovered.

Local Lipschitz continuity of minimizers of certain integrals of the
Calculus of Variations is obtained when the integrands are convex with
respect to the gradient variable, but are . In turn, these regularity results entail existence of minimizers of
variational problems with non-homogeneous integrands with
respect to the gradient variable. The -dependence, explicitly appearing
in the integrands, adds significant technical difficulties in the proof.

Regularity results for minimal configurations of variational problems involving both bulk and surface energies and subject to a volume constraint are established. The bulk energies are convex functions with -power growth, but are otherwise not subjected to any further structure conditions. For a minimal configuration (), Hölder continuity of the function is proved as well as partial regularity of the boundary of the minimal set . Moreover, full regularity of the boundary of the minimal set is obtained...

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