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Existence of varifold minimizers for the multiphase Canham–Helfrich functional

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Abstract

We address the minimization of the Canham–Helfrich functional in presence of multiple phases. The problem is inspired by the modelization of heterogeneous biological membranes, which may feature variable bending rigidities and spontaneous curvatures. With respect to previous contributions, no symmetry of the minimizers is here assumed. Correspondingly, the problem is reformulated and solved in the weaker frame of oriented curvature varifolds. We present a lower semicontinuity result and prove existence of single- and multiphase minimizers under area and enclosed-volume constrains. Additionally, we discuss regularity of minimizers and establish lower and upper diameter bounds.

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Acknowledgements

The authors gratefully acknowledge some valuable comments by Sascha Eichmann on a former version of the paper. Luca Lussardi is grateful to the kind hospitality of the University of Vienna where part of the work was performed. This work has been partially supported by the Vienna Science and Technology Fund (WWTF) through the Project MA14-009 and by the Austrian Science Fund (FWF) Projects F 65 and W 1245.

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Authors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Wien, Austria

    Katharina Brazda & Ulisse Stefanelli

  2. Dipartimento di Scienze Matematiche “G.L. Lagrange”, Politecnico di Torino, c.so Duca degli Abruzzi 24, 10129, Torino, Italy

    Luca Lussardi

  3. Vienna Research Platform on Accelerating Photoreaction Discovery, University of Vienna, Währingerstraße 17, 1090, Wien, Austria

    Ulisse Stefanelli

  4. Istituto di Matematica Applicata e Tecnologie Informatiche “E. Magenes”, via Ferrata 1, 27100, Pavia, Italy

    Ulisse Stefanelli

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  1. Katharina Brazda
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  2. Luca Lussardi
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  3. Ulisse Stefanelli
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Correspondence to Ulisse Stefanelli.

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Communicated by A. Malchiodi.

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Brazda, K., Lussardi, L. & Stefanelli, U. Existence of varifold minimizers for the multiphase Canham–Helfrich functional. Calc. Var. 59, 93 (2020). https://doi.org/10.1007/s00526-020-01759-9

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