JournalsaihpcVol. 21, No. 6pp. 839–880

Characterization and representation of the lower semicontinuous envelope of the elastica functional

  • G. Bellettini

    Dipartimento di Matematica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica, 00133 Roma, Italy
  • L. Mugnai

    Dipartimento di Matematica, Università di Pisa, via Buonarroti 2, 56127 Pisa, Italy
Characterization and representation of the lower semicontinuous envelope of the elastica functional cover
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Abstract

We characterize the lower semicontinuous envelope F\limits^{―} of the functional F(E):=∫∂E[1+|κ∂E|p] dH1, defined on boundaries of sets E⊂ℝ2, where κ∂E denotes the curvature of ∂E and p>1. Through a desingularization procedure, we find the domain of F\limits^{―} and its expression, by means of different representation formulas.

Résumé

On caractérise l’enveloppe semi-continue inférieure F\limits^{―} de la fonctionnelle F(E):=∫∂E[1+|κ∂E|p] dH1, définie sur la classe des frontières des domaines E⊂ℝ2, où κ∂E dénote la courbure de ∂E et p>1. Grâce à une méthode de désingularisation, on trouve le domaine de F\limits^{―} et son expression, à l’aide de différentes formules de représentation.

Cite this article

G. Bellettini, L. Mugnai, Characterization and representation of the lower semicontinuous envelope of the elastica functional. Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), no. 6, pp. 839–880

DOI 10.1016/J.ANIHPC.2004.01.001