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

Luciano Carbone; Riccardo De Arcangelis

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2001)

  • Volume: 12, Issue: 2, page 85-106
  • ISSN: 1120-6330

Abstract

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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 of open sets and less smooth functions. A suitable extension is constructed, and minimal sufficient conditions for its uniqueness are proposed. The results are applied to some examples in Calculus of Variations.

How to cite

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Carbone, Luciano, and De Arcangelis, Riccardo. "On the unique extension problem for functionals of the calculus of variations." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 12.2 (2001): 85-106. <http://eudml.org/doc/252367>.

@article{Carbone2001,
abstract = {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 $\mathbb\{R\}^\{n\}$ and every function in $C^\{\infty\} (\mathbb\{R\}^\{n\})$, 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 of open sets and less smooth functions. A suitable extension is constructed, and minimal sufficient conditions for its uniqueness are proposed. The results are applied to some examples in Calculus of Variations.},
author = {Carbone, Luciano, De Arcangelis, Riccardo},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Extension of functionals; Uniqueness; Lower semicontinuous envelopes; Inner regular envelopes; extension of functionals; uniqueness; lower semicontinuous envelopes; inner regular envelopes},
language = {eng},
month = {6},
number = {2},
pages = {85-106},
publisher = {Accademia Nazionale dei Lincei},
title = {On the unique extension problem for functionals of the calculus of variations},
url = {http://eudml.org/doc/252367},
volume = {12},
year = {2001},
}

TY - JOUR
AU - Carbone, Luciano
AU - De Arcangelis, Riccardo
TI - On the unique extension problem for functionals of the calculus of variations
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2001/6//
PB - Accademia Nazionale dei Lincei
VL - 12
IS - 2
SP - 85
EP - 106
AB - 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 $\mathbb{R}^{n}$ and every function in $C^{\infty} (\mathbb{R}^{n})$, 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 of open sets and less smooth functions. A suitable extension is constructed, and minimal sufficient conditions for its uniqueness are proposed. The results are applied to some examples in Calculus of Variations.
LA - eng
KW - Extension of functionals; Uniqueness; Lower semicontinuous envelopes; Inner regular envelopes; extension of functionals; uniqueness; lower semicontinuous envelopes; inner regular envelopes
UR - http://eudml.org/doc/252367
ER -

References

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Citations in EuDML Documents

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  1. Carmen Perugia, Omogenizzazione di problemi di tipo stazionario ed evolutivo in domini perforati e risultati di estensione unica nel Calcolo delle Variazioni
  2. Luciano Carbone, Doina Cioranescu, Riccardo De Arcangelis, Antonio Gaudiello, Homogenization of unbounded functionals and nonlinear elastomers. The case of the fixed constraints set
  3. Luisa Faella, Su alcuni problemi nell’omogeneizzazione e risultati di estensione unica nel calcolo delle variazioni
  4. Luciano Carbone, Doina Cioranescu, Riccardo De Arcangelis, Antonio Gaudiello, Homogenization of unbounded functionals and nonlinear elastomers. The case of the fixed constraints set

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