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### A derivation formula for convex integral functionals defined on $\text{BV}\left(\text{Ω}\right)$.

Journal of Convex Analysis

### Omogeneizzazione di funzionali debolmente quasi periodici

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Sia $f=f(x,z)$ quasiconvessa in $z$, quasiperiodica in $x$ nel senso di Besicovitch e soddisfi le disuguaglianze: $|z|^{p}\leq f(x,z)\leq\Lambda(1+|z|^{p}).$ Allora $f$ può essere omogeneizzata: esiste una funzione $\Psi$ che dipende solo da $z$ tale che i funzionali $\int_{\Omega}f\left(\frac{x}{\epsilon},Du(x)\right)\,dx\qquad u\in H^{1,p}(% \Omega;\mathbb{R}^{m})$ convergono, per $\epsilon$ tendente a $0$ (nel senso della $\Gamma$-convergenza) a $\int_{\Omega}\Psi(Du(x))\,dx.$ Inoltre si può dare una formula asintotica per $\Psi$.

### Omogeneizzazione di funzionali debolmente quasi periodici

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

Sia $f=f(x,z)$ quasiconvessa in $z$, quasiperiodica in $x$ nel senso di Besicovitch e soddisfi le disuguaglianze: $|z|^{p}\leq f(x,z)\leq\Lambda(1+|z|^{p}).$ Allora $f$ può essere omogeneizzata: esiste una funzione $\Psi$ che dipende solo da $z$ tale che i funzionali $\int_{\Omega}f\left(\frac{x}{\epsilon},Du(x)\right)\,dx\qquad u\in H^{1,p}(% \Omega;\mathbb{R}^{m})$ convergono, per $\epsilon$ tendente a $0$ (nel senso della $\Gamma$-convergenza) a $\int_{\Omega}\Psi(Du(x))\,dx.$ Inoltre si può dare una formula asintotica per $\Psi$.

### A variational model in image processing with focal points

ESAIM: Mathematical Modelling and Numerical Analysis

We propose a model for segmentation problems involving an energy concentrated on the vertices of an unknown polyhedral set, where the contours of the images to be recovered have preferred directions and focal points. We prove that such an energy is obtained as a -limit of functionals defined on sets with smooth boundary that involve curvature terms of the boundary. The minimizers of the limit functional are polygons with edges either parallel to some prescribed directions or pointing to some fixed...

### Multi-Scale Free-Discontinuity Problems with Soft Inclusions

Bollettino dell'Unione Matematica Italiana

### A compactness result for a second-order variational discrete model

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We analyze a nonlinear discrete scheme depending on second-order finite differences. This is the two-dimensional analog of a scheme which in one dimension approximates a free-discontinuity energy proposed by Blake and Zisserman as a higher-order correction of the Mumford and Shah functional. In two dimension we give a compactness result showing that the continuous problem approximating this difference scheme is still defined on special functions with bounded hessian, and we give an upper and a lower...

### Relaxation of elastic energies with free discontinuities and constraint on the strain

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

As a model for the energy of a brittle elastic body we consider an integral functional consisting of two parts: a volume one (the usual linearly elastic energy) which is quadratic in the strain, and a surface part, which is concentrated along the fractures (i.e. on the discontinuities of the displacement function) and whose density depends on the jump part of the strain. We study the problem of the lower semicontinuous envelope of such a functional under the assumptions that the surface energy density...

### A compactness result for a second-order variational discrete model

ESAIM: Mathematical Modelling and Numerical Analysis

We analyze a nonlinear discrete scheme depending on second-order finite differences. This is the two-dimensional analog of a scheme which in one dimension approximates a free-discontinuity energy proposed by Blake and Zisserman as a higher-order correction of the Mumford and Shah functional. In two dimension we give a compactness result showing that the continuous problem approximating this difference scheme is still defined on special functions...

### A-Quasiconvexity: Relaxation and Homogenization

ESAIM: Control, Optimisation and Calculus of Variations

Integral representation of relaxed energies and of -limits of functionals $\left(u,v\right)↦{\int }_{\Omega }f\left(x,u\left(x\right),v\left(x\right)\right)\phantom{\rule{0.166667em}{0ex}}dx$ 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.

### A relaxation result for energies defined on pairs set-function and applications

ESAIM: Control, Optimisation and Calculus of Variations

We consider, in an open subset of ${ℝ}^{N}$, energies depending on the perimeter of a subset $E\subset \Omega$ (or some equivalent surface integral) and on a function which is defined only on $\Omega \setminus E$. We compute the lower semicontinuous envelope of such energies. This relaxation has to take into account the fact that in the limit, the “holes” may collapse into a discontinuity of , whose surface will be counted twice in the relaxed energy. We discuss some situations where such energies appear, and give, as an application,...

### Homogenization of periodic multi-dimensional structures

Bollettino dell'Unione Matematica Italiana

Si studia il comportamento asintotico di una classe di funzionali integrali che possono dipendere da misure concentrate su strutture periodiche multidimensionali, quando tale periodo tende a 0. Il problema viene ambientato in spazi di Sobolev rispetto a misure periodiche. Si dimostra, sotto ipotesi generali, che un appropriato limite può venire definito su uno spazio di Sobolev usuale usando tecniche di $\mathrm{\Gamma }$-convergenza. Il limite viene espresso come un funzionale integrale il cui integrando è caratterizzato...

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