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Line energies for gradient vector fields in the plane

  • Luigi Ambrosio
  • Camillo De Lellis
  • Carlo Mantegazza
Original article

Abstract.

In this paper we study the singular perturbation of \(\int (1-|\nabla u|^2)^2\) by \(\varepsilon^2|\nabla^2u|^2\). This problem, which could be thought as the natural second order version of the classical singular perturbation of the potential energy \(\int (1-u^2)^2\) by \(\varepsilon^2|\nabla u|^2\), leads, as in the first order case, to energy concentration effects on hypersurfaces. In the two dimensional case we study the natural domain for the limiting energy and prove a compactness theorem in this class.

Keywords

Potential Energy Vector Field Energy Concentration Concentration Effect Dimensional Case 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Luigi Ambrosio
    • 1
  • Camillo De Lellis
    • 1
  • Carlo Mantegazza
    • 1
  1. 1.Scuola Normale Superiore, Piazza Cavalieri 7, I-56100 Pisa, Italy (e-mail: ambrosio@bibsns.sns.it / delellis@cibs.sns.it / mantegaz@cibs.sns.it) IT

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