Existence, regularity and structure of confined elasticae
ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 1, pp. 25-43.

We consider the problem of minimizing the bending or elastic energy among Jordan curves confined in a given open set Ω. We prove existence, regularity and some structural properties of minimizers. In particular, when Ω is convex we show that a minimizer is necessarily a convex curve. We also provide an example of a minimizer with self-intersections.

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2016073
Classification : 49J52, 49N60, 49Q10, 53A04
Mots clés : Minimization, confined curves, elastic energy, bending energy
Dayrens, François 1 ; Masnou, Simon 1 ; Novaga, Matteo 2

1 Institut Camille Jordan, Université Lyon 1, 43, Boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France.
2 Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy.
@article{COCV_2018__24_1_25_0,
     author = {Dayrens, Fran\c{c}ois and Masnou, Simon and Novaga, Matteo},
     title = {Existence, regularity and structure of confined elasticae},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {25--43},
     publisher = {EDP-Sciences},
     volume = {24},
     number = {1},
     year = {2018},
     doi = {10.1051/cocv/2016073},
     mrnumber = {3764132},
     zbl = {1397.49020},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2016073/}
}
TY  - JOUR
AU  - Dayrens, François
AU  - Masnou, Simon
AU  - Novaga, Matteo
TI  - Existence, regularity and structure of confined elasticae
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2018
SP  - 25
EP  - 43
VL  - 24
IS  - 1
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2016073/
DO  - 10.1051/cocv/2016073
LA  - en
ID  - COCV_2018__24_1_25_0
ER  - 
%0 Journal Article
%A Dayrens, François
%A Masnou, Simon
%A Novaga, Matteo
%T Existence, regularity and structure of confined elasticae
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2018
%P 25-43
%V 24
%N 1
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2016073/
%R 10.1051/cocv/2016073
%G en
%F COCV_2018__24_1_25_0
Dayrens, François; Masnou, Simon; Novaga, Matteo. Existence, regularity and structure of confined elasticae. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 1, pp. 25-43. doi : 10.1051/cocv/2016073. http://archive.numdam.org/articles/10.1051/cocv/2016073/

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000). | MR | Zbl

L. Ambrosio and S. Masnou, A direct variational approach to a problem arising in image reconstruction. Interfaces Free Bound. 5 (2003) 63–81. | DOI | MR | Zbl

G. Arreaga, R. Capovilla, C. Chryssomalakos and J. Guven, Area-constrained planar elastica. Phys. Rev. E 65 (2002). | DOI

T. Aubin, Some Nonlinear Problems in Riemannian Geometry. Springer Monographs in Mathematics. Springer (1998). | MR | Zbl

S. Avvakumov, O. Karpenkov and A. Sassinsky, Euler elasticae in the plane and the Whitney-Graustein theorem. Russian J. Math. Phys. 20 (2013) 257–267. | DOI | MR | Zbl

G. Bellettini, G. Dal Maso and M. Paolini, Semi-continuity and relaxation properties of a curvature depending functional in 2D. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 20 (1993) 247–297. | Numdam | MR | Zbl

G. Bellettini and L. Mugnai, Characterization and representation of the lower semicontinuous envelope of the elastica functional. Ann. Inst. Henri Poincaré 21 (2004) 839–880. | DOI | Numdam | MR | Zbl

G. Bellettini and L. Mugnai, A varifolds representation of the relaxed elastica functional. J. Convex Anal. 14 (2007) 543–564. | MR | Zbl

K. Bredies, T. Pock and B. Wirth, A convex, lower semi-continuous approximation of Euler’s elastica energy. SIAM J. Math. Anal. 47 (2015) 566–613. | DOI | MR | Zbl

E. Bretin, J.-O. Lachaud and E. Oudet, Regularization of discrete contour by Willmore energy. J. Math. Imaging Vis. 40 (2011) 214–229. | DOI | MR | Zbl

D. Bucur and A. Henrot, A new isoperimetric inequality for the elasticae. To appear in: J. Eur. Math. Soc. (2017). | MR

F. Cao, Y. Gousseau, S. Masnou and P. Pérez, Geometrically guided exemplar-based inpainting. SIAM J. Imaging Sci. 4 (2011) 1143–1179. | DOI | MR | Zbl

T. Chan, S.H. Kang and J. Shen, Euler’s elastica and curvature based inpaintings. SIAM J. Appl. Math. 63 (2002) 564–592. | MR | Zbl

G. Citti and A. Sarti, A cortical based model of perceptual completion in the roto-translation space. J. Math. Imaging Vis. 24 (2006) 307–326. | DOI | MR | Zbl

I. Coope, Curve interpolation with nonlinear spiral splines. IMA J. Numer. Anal. 13 (1992) 327–341. | DOI | MR | Zbl

P. Dondl, L. Mugnai and M. Röger, Confined elastic curves. SIAM J. Appl. Math. 71 (2011) 2205–2226. | DOI | MR | Zbl

P.W. Dondl, A. Lemenant and S. Wojtowytsch, Phase field models for thin elastic structures with topological constraint. Arch. Ration. Mech. Anal. 223 (2017) 693–736. | DOI | MR | Zbl

S. Esedoglu and J. Shen, Digital image inpainting by the Mumford-Shah-Euler image model. European J. Appl. Math. 13 (2002) 353–370. | DOI | MR | Zbl

L.C. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press (1992). | MR | Zbl

V. Ferone, B. Kawohl and C. Nitsch, The elastica problem under area constraint. Math. Ann. 365 (2016) 987–1015. | DOI | MR | Zbl

G. Forsythe and E. Lee, Variational study of nonlinear spline curves. SIAM Rev. 15 (1973) 120–133. | DOI | MR | Zbl

M. Giaquinta and G. Modica, Mathematical Analysis, Fundations and Advanced Techniques for Functions of Several Variables. Birkhauser (2012). | Zbl

E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities. Lecture notes. American Mathematical Society (2000). | Zbl

B. Horn, The curve of least energy. ACM Trans. Math. Softw. 9 (1983) 441–460. | DOI | MR | Zbl

N. Koiso, Elastica in a Riemannian submanifold. Osaka J. Math. 29 (1992) 539–543. | MR | Zbl

J. Langer and D. Singer, Knotted elastic curves in R 3 . Kangwon-Kyunggi Math. J. 5 (1984) 113–119.

J. Langer and D. Singer, The total squared curvature of closed curves. J. Differ. Geom. 20 (1984) 1–22. | DOI | MR | Zbl

J. Langer and D. Singer, Curve straightening and a minimax argument for closed elastic curves. Topology 24 (1985) 75–88. | DOI | MR | Zbl

A. Linnér, Some properties of the curve straightening flow in the plane. Trans. Amer. Math. Soc. 314 (1989) 605–618. | DOI | MR | Zbl

S. Masnou and J.M. Morel, On a variational theory of image amodal completion. Rend. Semin. Mat. Univ. Padova 116 (2006) 211–252. | Numdam | MR | Zbl

T. Miura, Singular perturbation by bending for an adhesive obstacle problem. To appear in Calc. Var. Partial Differ. Equ. (2016). | MR

D. Mumford, Elastica and computer vision. In Algebraic Geometry and its Applications, edited by C. Bajaj. Springer Verlag, New York (1994) 491–506. | MR | Zbl

M. Nitzberg and D. Mumford, The 2.1-D Sketch. In Proc. of 3rd Int. Conf. on Computer Vision, Osaka, Japan (1990) 138–144.

M. Novaga and S. Okabe, Curve shortening-straightening flow for non-closed planar curves with infinite length. J. Differ. Eqs. 256 (2014) 1093–1132. | DOI | MR | Zbl

S. Okabe, The motion of elastic planar closed curves under the area preserving condition. Ind. Univ. Math. J. 56 (2007) 1871–1912. | DOI | MR | Zbl

N. Olischläger and M. Rumpf, Two step time discretization of Willmore flow. Lect. Notes Comput. Sci. 5654 (2009) 278–292. | DOI | Zbl

Y. Sachkov, Maxwell strata in the Euler elastic problem. J. Dyn. Control Syst. 14 (2008)169–234. | MR | Zbl

Y. Sachkov, Closed Euler elasticae. Proc. of Steklov Inst. Math. 278 (2012) 218–232. | DOI | MR | Zbl

T. Schoenemann, F. Kahl, S. Masnou and D. Cremers, A linear framework for region-based image segmentation and inpainting involving curvature penalization. Inter. J. Comput. Vision 99 (2012) 53–68. | DOI | MR | Zbl

T. Schoenemann, S. Masnou and D. Cremers, The elastic ratio: Introducing curvature into ratio-based globally optimal image segmentation. IEEE Trans. Image Proc. 20 (2011) 2565–2581. | DOI | MR | Zbl

J. Ulén, P. Strandmark and F. Kahl, Shortest paths with higher-order regularization. IEEE Trans. Pattern Anal. Mach. Intel. (2015).

Cité par Sources :