Abstract

Under suitable regularity assumptions, the pp-elastic energy of a planar set E⊂R2E\subset\mathbb{R}^2 is defined as Fp(E)=∫∂E1+∣k∂E∣p  dH1,\mathcal{F}_p(E)=\int_{\partial E} 1 + |k_{\partial E}|^p \,\, d\mathcal{H}^1, where k∂Ek_{\partial E} is the curvature of the boundary ∂E\partial E. In this work we use a varifold approach to investigate this energy, that can be well defined on varifolds with curvature. First we show new tools for the study of 11-dimensional curvature varifolds, such as existence and uniform bounds on the density of varifolds with finite elastic energy. Then we characterize a new notion of L1L^1-relaxation of this energy by extending the definition of regular sets by an intrinsic varifold perspective, also comparing this relaxation with the classical one of G. Bellettini and L. Mugnai [Characterization and representation of the lower semicontinuous envelope of the elastica functional, Annales de l'Institut Henri Poincaré (C), Non Linear Analysis 21(6) (2004) 839–880; A varifolds representation of the relaxed elastica functional, J. Convex Analysis 14(3) (2007) 543–564]. Finally we discuss an application to the inpainting problem, examples and qualitative properties of sets with finite relaxed energy.

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Keywords

  • Curvature varifolds
  • p-elastic energy
  • relaxation

Mathematics Subject Classification

49Q15, 49Q20, 49Q10, 53A07

References

  1. L. Ambrosio, N. Fusco, D. Pallara: Functions of Bounded Variation and Free Discontinuity Problems, Oxford Science Publications, Oxford (2000).https://doi.org/10.1093/oso/9780198502456.001.0001Google Scholar
  2. L. Ambrosio, S. Masnou: A direct variational approach to a problem arising in image reconstruction, Interfaces Free Boundaries 5 (2003) 63–81.https://doi.org/10.4171/ifb/72Google Scholar
  3. G. Bellettini, G. Dal Maso, M. Paolini: Semicontinuity and relaxation properties of a curvature depending functional in 2D, Ann. Scuola Normale Sup. Pisa, Classe di Scienze 20(2) (1993) 247–297.Google Scholar
  4. G. Bellettini, L. Mugnai: Characterization and representation of the lower semicontinuous envelope of the elastica functional, Ann. Inst. H. Poincaré, Anal. Non Linéaire 21(6) (2004) 839–880.https://doi.org/10.1016/j.anihpc.2004.01.001Google Scholar
  5. G. Bellettini, L. Mugnai: A varifolds representation of the relaxed elastica functional, J. Convex Analysis 14(3) (2007) 543–564.https://doi.org/10.68381/jca14033Google Scholar
  6. G. Bellettini, M. Paolini: Variational properties of an image segmentation functional depending on contours curvature, Adv. Math. Science Appl., Gakkötosho, Tokyo 5(2) (1995) 681–715.Google Scholar
  7. M. Bertalmío, V. Caselles, S. Masnou, G. Sapiro: Inpainting, Computer Vision, Springer, Boston (2016).Google Scholar
  8. A. Dall’Acqua, M. Novaga, A. Pluda: Minimal elastic networks, Indiana Univ. Math. J., to appear.Google Scholar
  9. G. P. Leonardi, S. Masnou: Locality of the mean curvature of rectifiable varifolds, Adv. Calc. Var. 2(1) (2009) 17–42.https://doi.org/10.1515/ACV.2009.002Google Scholar
  10. S. Luckhaus, T. Sturzenhecker: Implicit time discretization for the mean curvature flow equation, Calc. Var 3 (1995) 253–271.https://doi.org/10.1007/BF01205007Google Scholar
  11. S. Masnou, G. Nardi: A coarea-type formula for the relaxation of a generalized elastica functional, J. Convex Analysis 20(3) (2013) 617–653.https://doi.org/10.68381/jca20036Google Scholar
  12. S. Masnou, G. Nardi: Gradient Young measures, varifolds, and a generalized Willmore functional, Adv. Calc. Var. 6(4) (2013) 433–482.https://doi.org/10.1515/acv-2011-0014Google Scholar
  13. U. Menne: Weakly differentiable functions on varifolds, Indiana Univ. Math. J. 65(3) (2016) 977–1088.https://doi.org/10.1512/iumj.2016.65.5829Google Scholar
  14. J. W. Milnor: Topology from the Differentiable Viewpoint, The University Press of Virginia, Charlottesville (1965).Google Scholar
  15. S. Okabe, P. Pozzi, G. Wheeler: A gradient flow for the p-elastic energy defined on closed planar curves, Math. Annalen (2019), arXiv: 1811.06608 (2018).Google Scholar
  16. O. Ore: Theory of Graphs, Colloquium Publications 38, American Mathematical Society, Providence (1962).https://doi.org/10.1090/coll/038Google Scholar
  17. L. Simon: Lectures on Geometric Measure Theory, Proc. Centre Math. Analysis, Australian National University, Canberra (1984).Google Scholar
  18. L. Simon: Existence of surfaces minimizing the Willmore functional, Comm. Analysis Geometry 1 (1993) 281–326.https://doi.org/10.4310/CAG.1993.v1.n2.a4Google Scholar

Suggested citation

M. Pozzetta. “A Varifold Perspective on the p-Elastic Energy of Planar Sets.” Journal of Convex Analysis 27 (2020), No. 3, 845–879. https://doi.org/10.68381/jca27043

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Published by Heldermann Verlag, 2020. Rights now held by Banach Press.