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Minimal surfaces with elastic and partially elastic boundary

Part of: Classical differential geometry Variational problems in infinite-dimensional spaces

Published online by Cambridge University Press:  20 August 2020

Bennett Palmer  and
Álvaro Pámpano
Bennett Palmer
Affiliation:
Department of Mathematics, Idaho State University, Pocatello, ID83209, USA ([email protected])
Álvaro Pámpano
Affiliation:
Department of Mathematics, University of the Basque Country, Bilbao, Spain ([email protected])
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Abstract

We study equilibrium surfaces for an energy which is a linear combination of the area and a second term which measures the bending and twisting of the boundary. Specifically, the twisting energy is given by the twisting of the Darboux frame. This energy is a modification of the Euler–Plateau functional considered by Giomi and Mahadevan (2012, Proc. R. Soc. A 468, 1851–1864), and a natural special case of the Kirchhoff–Plateau energy considered by Biria and Fried (2014, Proc. R. Soc. A 470, 20140368; 2015, Int. J. Eng. Sci. 94, 86–102).

Keywords

bending energiesDarboux frameminimal surfaces

MSC classification

Primary: 58E12: Applications to minimal surfaces (problems in two independent variables)
Secondary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature 53A04: Curves in Euclidean space
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 4 , August 2021 , pp. 1225 - 1246
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Copyright
Copyright © The Author(s) 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Alt, H. W.. Die existenz eines minimalflache mit freimen rand vorgeschriebrener lange. Arch. Ration. Mech. Anal. 51 (1973), 304–320.CrossRefGoogle Scholar
Arreaga, G., Capovilla, R., Chryssomalakos, C. and Guven, J.. Area-constrained planar elastica. Phys. Rev. E. 65 (2002), 031801.CrossRefGoogle ScholarPubMed
Bevilacqua, G., Lussardi, L. and Marzocchi, A.. Soap film spanning an elastic link. Quart. Appl. Math. 77 (2019), 507–523.CrossRefGoogle Scholar
Biria, A. and Fried, E.. Buckling of a soap film spanning a flexible loop resistant to bending and twisting. Proc. R. Soc. A 470 (2014), 20140368.CrossRefGoogle Scholar
Biria, A. and Fried, E.. Theoretical and experimental study of the stability of a soap film spanning a flexible loop. Int. J. Eng. Sci. 94 (2015), 86–102.CrossRefGoogle Scholar
Evans, K. E., Nkansah, M. A., Hutchinson, I. J. and Rogers, S. C.. Molecular network design. Nature 353 (1991), 124.CrossRefGoogle Scholar
Feynman, R. P.. Feynman Lectures on Physics. Mainly Electromagnetism and Matter, vol. 2 (Boston, USA: Addison-Wesley, 1964).Google Scholar
Giusteri, G. G., Lussardi, L. and Fried, E.. Solution of the Kirchhoff–Plateau problem. J. Nonlinear Sci. 27 (2017), 1043–1063.CrossRefGoogle ScholarPubMed
Giomi, L. and Mahadevan, L.. Minimal surfaces bounded by elastic lines. Proc. R. Soc. A 468 (2012), 1851–1864.CrossRefGoogle Scholar
Giusteri, G. G., Franceschini, P. and Fried, E.. Instability paths in the Kirchhoff–Plateau problem. J. Nonlinear Sci. 26 (2016), 1097–1132.CrossRefGoogle Scholar
Guven, J., Valencia, D. M. and Vázquez-Montejo, P.. Environmental bias and elastic curves on surfaces. J. Phys. A: Math. Theor. 47 (2014), 355201.CrossRefGoogle Scholar
Hille, E.. Ordinary Differential Equations in the Complex Domain (New York: Courier Corporation, 1997).Google Scholar
Hopf, H.. Differential Geometry in the Large. Seminar Lectures New York University 1946 and Stanford University 1956, vol. 1000 (Berlin: Springer, 2003).Google Scholar
Langer, J. and Singer, D. A.. Lagrangian aspects of the Kirchhoff elastic rod. SIAM Rev. 38 (1986), 605–618.CrossRefGoogle Scholar
Lawson, H. B.. Some intrinsic characterizations of minimal surfaces. J. Anal. Math. 24 (1971), 151–161.CrossRefGoogle Scholar
Love, A. E. H.. A Treatise on the Mathematical Theory of Elasticity, 4th edn (New York: Dover Publications, 1944).Google Scholar
Marshall, W. F.. Differential geometry meets the cell. Cell 154 (2013), 265–266.CrossRefGoogle ScholarPubMed
Nitsche, J. C.. Stationary partitioning of convex bodies. Arch. Ration. Mech. Anal. 89 (1985), 1–19.CrossRefGoogle Scholar
Terasaki, M., Shemesh, T., Kasthuri, N., Klemm, R. W., Schalek, R., Hayworth, K. J., Hand, A. R., Yankova, M., Huber, G., Lichtman, J. W., Rapoport, T. A. and Kozlov, M. M.. Stacked endoplasmic reticulum sheets are connected by helicoidal membrane motifs. Cell 154 (2013), 285–296.CrossRefGoogle ScholarPubMed