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Elvira Zappale

Publications and source records attributed to Elvira Zappale.

51 records · Page 3Linked to original sources
arXiv2019

A relaxation result in the vectorial setting and $L^p$-approximation for $L^\infty$-functionals

We provide relaxation for not lower semicontinuous supremal functionals of the type $W^{1,\infty}(Ω;\mathbb R^d) \ni u \mapsto\supess_{ x \in Ω}f(\nabla u(x))$ in the vectorial case, where $Ω\subset \mathbb R^N$ is a Lipschitz, bounded open set, and $f$ is level convex. The connection with indicator functionals is also enlightened, thus extending previous lower semicontinuity results in that framework. Finally we discuss the $L^p$-approximation of supremal functionals, with non-negative, coercive densities $f=f(x,ξ)$, which are only $Ł^N \otimes \B_{d \times N}$-measurable.

math.OC↗
arXiv2017

Relaxation for an optimal design problem with nonstandard growth

In this paper we investigate the possibility of obtaining a measure representation for function- als arising in the context of optimal design problems under non-standard growth conditions and perimeter penalization. Applications to modelling of strings are also provided.

math.AP↗
arXiv2017

Bending-torsion moments in thin multi-structures in the context of nonlinear elasticity

Here, we address a dimension-reduction problem in the context of nonlinear elasticity where the applied external surface forces induce bending-torsion moments. The underlying body is a multi-structure in $\mathbb{R}^3$ consisting of a thin tube-shaped domain placed upon a thin plate-shaped domain. The problem involves two small parameters, the radius of the cross-section of the tube-shaped domain and the thickness of the plate-shaped domain. We characterize the different limit models, including the limit junction condition, in the membrane-string regime according to the ratio between these two parameters as they converge to zero.

math.AP↗
arXiv2016

Optimal Design of Fractured Media with Prescribed Macroscopic Strain

In this work we consider an optimal design problem for two-component fractured media for which a macroscopic strain is prescribed. Within the framework of structured deformations, we derive an integral representation for the relaxed energy functional. We start from an energy functional accounting for bulk and surface contributions coming from both constituents of the material; the relaxed energy densities, obtained via a blow-up method, are determined by a delicate interplay between the optimization of sharp interfaces and the diffusion of microcracks. This model has the far-reaching perspective to incorporate elements of plasticity in optimal design of composite media.

math.OC↗
arXiv2014

Relaxation for an optimal design problem with linear growth and perimeter penalization

The paper is devoted to the relaxation and integral representation in the space of functions of bounded variation for an integral energy arising from optimal design problems. The presence of a perimeter penalization is also considered in order to avoid non existence of admissible solutions, besides this leads to an interaction in the limit energy. Also more general models have been taken into account.

math.FA↗
arXiv2013

Existence of Minimizers for Non-Level Convex Supremal Functionals

The paper is devoted to determine necessary and sufficient conditions for existence of solutions to the problem ${\rm inf}{{\rm ess sup}_{x \in Ω} f(\nabla u(x)): u \in u_0 + W^{1,\infty}_0(Ω)}$ when the supremand $f$ is not necessarily level convex. These conditions are obtained through a comparison with the related level convex problem and are written in terms of a differential inclusion involving the boundary datum. Several conditions of convexity for the supremand $f$ are also investigated.

math.OC↗
arXiv2012

Dimension reduction for $-Δ_1$

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.

math.AP↗
arXiv2012

Lower semicontinuous envelopes in $w^{1,1}\times l^p$

It is studied the lower semicontinuity of functionals of the type $\int_Ωf(x,u,v, \nabla u)dx$ with respect to the $(W^{1,1}\times L^p)$-weak \ast topology. Moreover in absence of lower semicontinuity, it is also provided an integral representation in $W^{1,1} \times L^p$ for the lower semicontinuous envelope.

math.AP↗
arXiv2009

Some sufficient conditions for lower semicontinuity in SBD and applications to minimum problems of Fracture Mechanics

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.

math.AP↗
arXiv2008

Dimensional reduction for energies with linear growth involving the bending moment

A $Γ$-convergence analysis is used to perform a 3D-2D dimension reduction of variational problems with linear growth. The adopted scaling gives rise to a nonlinear membrane model which, because of the presence of higher order external loadings inducing a bending moment, may depend on the average in the transverse direction of a Cosserat vector field, as well as on the deformation of the mid-plane. The assumption of linear growth on the energy leads to an asymptotic analysis in the spaces of measures and of functions with bounded variation.

math.AP↗