Abstract
We present rigorous analysis of Lagrange multipliers arising in the variational computations of a membrane’s energy functional. Through the convergence analysis, it is found that the local Lagrange multiplier is sensitive to the properties of membranes. Unlike the local Lagrange multiplier, the bulk liquid incompressibility condition via the global Lagrangian field acts as an additional boundary input rather than a constraint, in the perspective of the surrounding membrane. With the enhanced understanding of Lagrange multiplier fields, a membrane–substrate interaction problem is considered where singularities are present at the cross section of the substrate. A viable analytical solution is obtained under the modified boundary conditions. In addition, a complete analysis of necessary boundary condition in the case of flat surfaces is presented within the prescription of superposed incremental deformations.
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Kim, C.I. A discussion on the mechanics of lipid membranes: Lagrange multipliers and a singular substrate. Z. Angew. Math. Phys. 68, 84 (2017). https://doi.org/10.1007/s00033-017-0825-5
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DOI: https://doi.org/10.1007/s00033-017-0825-5