arXiv subjects
Publications and source records attributed to Giuliano Gargiulo.
In this note we present a sufficient condition ensuring lower semicontinuity for nonlocal supremal functionals of the type $$W^{1,\infty}(Ω;\mathbb R^d)\ni u \mapsto \sup{\rm ess}_{(x,y)\in Ω} W(x,y, \nabla u(x),\nabla u(y)),$$ where $Ω$ is a bounded open subset of $\mathbb R^N$ and $W:Ω\times Ω\times \mathbb R^{d \times N}\times \mathbb R^{d \times N} \to \mathbb R$.
The aim of this note consists of the introduction of unfolding operator in the Orlicz setting.
The $Γ$-limit of a family of functionals $u\mapsto \int_{Ω}f\left( \frac{x}{\varepsilon },\frac{x}{\varepsilon ^{2}},D^{s}u\right) dx$ is obtained for $s=1,2$ and when the integrand $f=f\left( y,z,v\right) $ is a continous function, periodic in $y$ and $z$ and convex with respect to $v$ with nonstandard growth. The reiterated two-scale limits of second order derivative are characterized in this setting.
A 3D-2D dimension reduction for $-Δ_1$ is obtained. A power law approximation from $-Δ_p$ as $p \to 1$ in terms of $Γ$- convergence, duality and asymptotics for least gradient functions has also been provided.
We provide some lower semicontinuity results in the space of special functions of bounded deformation for energies of the type $$ %\int_Ø{1/2}({\mathbb C} \E u, \E u)dx + \int_{J_{u}} Θ(u^+, u^-, ν_{u})d \H^{N-1} \enspace, \enspace [u]\cdot ν_u \geq 0 \enspace {\cal H}^{N-1}-\hbox{a. e. on}J_u, $$ and give some examples and applications to minimum problems. \noindent Keywords: Lower semicontinuity, fracture, special functions of bounded deformation, joint convexity, $BV$-ellipticity.