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

Accepted:

DOI: 10.1051/cocv/2016073

Keywords: Minimization, confined curves, elastic energy, bending energy

^{1}; Masnou, Simon

^{1}; Novaga, Matteo

^{2}

@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, Volume 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

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

and ,Area-constrained planar elastica. Phys. Rev. E 65 (2002). | DOI

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

Euler elasticae in the plane and the Whitney-Graustein theorem. Russian J. Math. Phys. 20 (2013) 257–267. | DOI | MR | Zbl

, and ,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

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

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

and ,A convex, lower semi-continuous approximation of Euler’s elastica energy. SIAM J. Math. Anal. 47 (2015) 566–613. | DOI | MR | Zbl

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

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

Geometrically guided exemplar-based inpainting. SIAM J. Imaging Sci. 4 (2011) 1143–1179. | DOI | MR | Zbl

, , and ,Euler’s elastica and curvature based inpaintings. SIAM J. Appl. Math. 63 (2002) 564–592. | MR | Zbl

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

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

,Confined elastic curves. SIAM J. Appl. Math. 71 (2011) 2205–2226. | DOI | MR | Zbl

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

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

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

The elastica problem under area constraint. Math. Ann. 365 (2016) 987–1015. | DOI | MR | Zbl

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

and ,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

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

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

,Knotted elastic curves in ${R}^{3}$. Kangwon-Kyunggi Math. J. 5 (1984) 113–119.

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

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

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

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

and ,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.

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

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

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

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

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

,A linear framework for region-based image segmentation and inpainting involving curvature penalization. Inter. J. Comput. Vision 99 (2012) 53–68. | DOI | MR | Zbl

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

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

*Cited by Sources: *