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August  2022, 21(8): 2819-2829. doi: 10.3934/cpaa.2022074

Minimisers of Helfrich functional for surfaces of revolution

Department of Mathematical Sciences, Tsinghua University, China

*Corresponding author

Received  January 2022 Revised  March 2022 Published  August 2022 Early access  April 2022

Fund Project: This work was supported by NSFC 12141103

In this paper we investigate the existence and the properties for the minimisers of a special Helfrich functional for surfaces of revolution with Dirichlet boundary value conditions. Removing the even restriction for the admissible functions in [5], we prove that the minimiser is even and smooth, the minimal increases as the boundary value increases, and the minimiser is no less than the boundary value which answers an open question in [5] partly. We also obtain the existence and regularity for (general) Helfrich functional when the boundary value is large.

Citation: Huaiyu Jian, Hongbo Zeng. Minimisers of Helfrich functional for surfaces of revolution. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2819-2829. doi: 10.3934/cpaa.2022074
References:
[1]

K. Brazda, L. Lussardi and U. Stefanelli, Existence of varifold minimizers for the multiphase Canham-Helfrich functional, Calc Var. Partial Differ. Equ., 59 (2020), 26 pp. doi: 10.1007/s00526-020-01759-9.

[2]

P. Canham, The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell, J. Theor. Biol., 26 (1970), 26-61. 

[3]

R. ChoksiM. Morandotti and M. Veneroni, Global minimizers for axisymmetric multiphase membranes, ESAIM, Control, Optimi. Calc. Var., 19 (1970), 1014-1029.  doi: 10.1051/cocv/2012042.

[4]

R. Choksi and M. Veneroni, Global minimizers for the doubly-constrained Helfrich energy: the axisymmetric case, Calc. Var. Partial Differ. Equ., 48 (2013), 337-366.  doi: 10.1007/s00526-012-0553-9.

[5]

K. Deckelnick, M. Doemeland and H. Grunau, Boundary value problems for a special Helfrich functional for surfaces of revolution: existence and asymptotic behaviour, Calc. Var. Partial Differ. Equ., 60 (2021), 31 pp. doi: 10.1007/s00526-020-01875-6.

[6]

K. DeckelnickH. Grunau and M. Roger, Minimising a relaxed Willmore functional for graphs subject to boundary conditions, Interf. Free Bound., 19 (2017), 109-140.  doi: 10.4171/IFB/378.

[7]

S. Eichmann and H. Grunau, Existence for Willmore surfaces of revolution satisfying non-symmetric Dirichlet boundary conditions, Adv. Calc. Var., 12 (2019), 333-361.  doi: 10.1515/acv-2016-0038.

[8]

S. Eichmann and A. Koeller, Symmetry for Willmore Surfaces of Revolution, J. Geom. Anal., 27 (2016), 1-25. 

[9]

H. Frieske and M. Mahnig, Elastic properties of lipid bilayers: theory and possible experiments, Zeitschrift Fur Naturforschung Section C-a Journal of Biosciences, 28 (1973), 693-703. 

[10] M. Giaquinta and S. Hildebrandt, Calculus of Variations I, Oxford University Press, 1996. 
[11]

H. Jian and H. Zeng, Existence and uniqueness for variational problem from progressive lens design, Front. Math. China, 15 (2020), 491-505.  doi: 10.1007/s11464-020-0845-x.

[12]

E. Kuwert and R. Schatzle, Closed surfaces with bounds on their Willmore energy, Ann. Della Scuola Norm. Super. Pisa Classe Di Sci., 11 (2010), 605-634. 

[13]

P. Li and S. Yau, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math., 69 (1982), 269-291.  doi: 10.1007/BF01399507.

[14]

Y. Li, Some remarks on Willmore surfaces embedded in $R^3$, J. Geom. Anal., 26 (2016), 2411-2424.  doi: 10.1007/s12220-015-9631-5.

[15]

F. Marques and A. Neves, Min-Max theorey and the Willmore conjecture, Ann. Math., (2014), 683–782. doi: 10.4007/annals.2014.179.2.6.

[16]

B. Matthias and K. Ernst, Existence of minimizing Willmore surfaces of prescribed genus, Int. Math. Res. Notices, 10 (2003), 553-576. 

[17]

R. Schatzle, The Willmore boundary problem, Calc. Var. Partial Differ. Equ., 37 (2010), 275-302.  doi: 10.1007/s00526-009-0244-3.

[18]

S. Scholtes, Elastic catenoids, Int. Math. J. Anal. Appl., 31 (2011), 125-143.  doi: 10.1524/anly.2011.1088.

[19]

Y. Xie and Z. Ou-Yang, Helfrich Theory of Biomembranes, Modern Phys. Lett. B, 6 (1992), 917-933. 

show all references

References:
[1]

K. Brazda, L. Lussardi and U. Stefanelli, Existence of varifold minimizers for the multiphase Canham-Helfrich functional, Calc Var. Partial Differ. Equ., 59 (2020), 26 pp. doi: 10.1007/s00526-020-01759-9.

[2]

P. Canham, The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell, J. Theor. Biol., 26 (1970), 26-61. 

[3]

R. ChoksiM. Morandotti and M. Veneroni, Global minimizers for axisymmetric multiphase membranes, ESAIM, Control, Optimi. Calc. Var., 19 (1970), 1014-1029.  doi: 10.1051/cocv/2012042.

[4]

R. Choksi and M. Veneroni, Global minimizers for the doubly-constrained Helfrich energy: the axisymmetric case, Calc. Var. Partial Differ. Equ., 48 (2013), 337-366.  doi: 10.1007/s00526-012-0553-9.

[5]

K. Deckelnick, M. Doemeland and H. Grunau, Boundary value problems for a special Helfrich functional for surfaces of revolution: existence and asymptotic behaviour, Calc. Var. Partial Differ. Equ., 60 (2021), 31 pp. doi: 10.1007/s00526-020-01875-6.

[6]

K. DeckelnickH. Grunau and M. Roger, Minimising a relaxed Willmore functional for graphs subject to boundary conditions, Interf. Free Bound., 19 (2017), 109-140.  doi: 10.4171/IFB/378.

[7]

S. Eichmann and H. Grunau, Existence for Willmore surfaces of revolution satisfying non-symmetric Dirichlet boundary conditions, Adv. Calc. Var., 12 (2019), 333-361.  doi: 10.1515/acv-2016-0038.

[8]

S. Eichmann and A. Koeller, Symmetry for Willmore Surfaces of Revolution, J. Geom. Anal., 27 (2016), 1-25. 

[9]

H. Frieske and M. Mahnig, Elastic properties of lipid bilayers: theory and possible experiments, Zeitschrift Fur Naturforschung Section C-a Journal of Biosciences, 28 (1973), 693-703. 

[10] M. Giaquinta and S. Hildebrandt, Calculus of Variations I, Oxford University Press, 1996. 
[11]

H. Jian and H. Zeng, Existence and uniqueness for variational problem from progressive lens design, Front. Math. China, 15 (2020), 491-505.  doi: 10.1007/s11464-020-0845-x.

[12]

E. Kuwert and R. Schatzle, Closed surfaces with bounds on their Willmore energy, Ann. Della Scuola Norm. Super. Pisa Classe Di Sci., 11 (2010), 605-634. 

[13]

P. Li and S. Yau, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math., 69 (1982), 269-291.  doi: 10.1007/BF01399507.

[14]

Y. Li, Some remarks on Willmore surfaces embedded in $R^3$, J. Geom. Anal., 26 (2016), 2411-2424.  doi: 10.1007/s12220-015-9631-5.

[15]

F. Marques and A. Neves, Min-Max theorey and the Willmore conjecture, Ann. Math., (2014), 683–782. doi: 10.4007/annals.2014.179.2.6.

[16]

B. Matthias and K. Ernst, Existence of minimizing Willmore surfaces of prescribed genus, Int. Math. Res. Notices, 10 (2003), 553-576. 

[17]

R. Schatzle, The Willmore boundary problem, Calc. Var. Partial Differ. Equ., 37 (2010), 275-302.  doi: 10.1007/s00526-009-0244-3.

[18]

S. Scholtes, Elastic catenoids, Int. Math. J. Anal. Appl., 31 (2011), 125-143.  doi: 10.1524/anly.2011.1088.

[19]

Y. Xie and Z. Ou-Yang, Helfrich Theory of Biomembranes, Modern Phys. Lett. B, 6 (1992), 917-933. 

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