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On the unique extension problem for functionals of the calculus of variations

Luciano CarboneRiccardo De Arcangelis — 2001

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

By drawing inspiration from the treatment of the non parametric area problem, an abstract functional is considered, defined for every open set in a given class of open subsets of R n and every function in C R n , and verifying suitable assumptions of measure theoretic type, of invariance, convexity, and lower semicontinuity. The problem is discussed of the possibility of extending it, and of the uniqueness of such extension, to a functional verifying analogous properties, but defined in wider families...

On integral representation, relaxation and homogenization for unbounded functionals

Luciano CarboneRiccardo De Arcangelis — 1997

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

A theory of integral representation, relaxation and homogenization for some types of variational functionals taking extended real values and possibly being not finite also on large classes of regular functions is presented. Some applications to gradient constrained relaxation and homogenization problems are given.

Homogenization of unbounded functionals and nonlinear elastomers. The case of the fixed constraints set

Luciano CarboneDoina CioranescuRiccardo De ArcangelisAntonio Gaudiello — 2004

ESAIM: Control, Optimisation and Calculus of Variations

The paper is a continuation of a previous work of the same authors dealing with homogenization processes for some energies of integral type arising in the modeling of rubber-like elastomers. The previous paper took into account the general case of the homogenization of energies in presence of pointwise oscillating constraints on the admissible deformations. In the present paper homogenization processes are treated in the particular case of fixed constraints set, in which minimal coerciveness hypotheses...

Homogenization of unbounded functionals and nonlinear elastomers. The case of the fixed constraints set

Luciano CarboneDoina CioranescuRiccardo De ArcangelisAntonio Gaudiello — 2010

ESAIM: Control, Optimisation and Calculus of Variations

The paper is a continuation of a previous work of the same authors dealing with homogenization processes for some energies of integral type arising in the modeling of rubber-like elastomers. The previous paper took into account the general case of the homogenization of energies in presence of pointwise oscillating constraints on the admissible deformations. In the present paper homogenization processes are treated in the particular case of fixed constraints set, in which minimal coerciveness hypotheses...

Homogenization of Neumann problems for unbounded integral functionals

Luciano CarboneAntonio Corbo EspositoRiccardo De Arcangelis — 1999

Bollettino dell'Unione Matematica Italiana

Si studia l'omogeneizzazione di problemi di tipo Neumann per funzionali integrali del Calcolo delle Variazioni definiti su funzioni soggette a vincoli puntuali di tipo oscillante sul gradiente, in ipotesi minimali sui vincoli. I risultati ottenuti sono dedotti mediante l'introduzione di nuove tecniche di Γ -convergenza, unitamente ad un risultato di ricostruzione per funzioni affini a tratti, che permettono la dimostrazione di un teorema generale di omogeneizzazione per funzionali integrali a valori...

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