# American Institute of Mathematical Sciences

January  2021, 14(1): 331-351. doi: 10.3934/dcdss.2020325

## Existence of weak solutions for a sharp interface model for phase separation on biological membranes

 Faculty of Mathematics, University of Regensburg, 93040 Regensburg, Germany

* Corresponding author: Helmut Abels

Received  March 2019 Revised  September 2019 Published  April 2020

We prove existence of weak solutions of a Mullins-Sekerka equation on a surface that is coupled to diffusion equations in a bulk domain and on the boundary. This model arises as a sharp interface limit of a phase field model to describe the formation of liqid rafts on a cell membrane. The solutions are constructed with the aid of an implicit time discretization and tools from geometric measure theory to pass to the limit.

Citation: Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325
##### References:
 [1] H. Abels and J. Kampmann, On the sharp interface limit of a model for phase separation on biological membranes, Preprint, arXiv: 1811.12489, 2018. Google Scholar [2] H. Abels and M. Röger, Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous, incompressible fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2403-2424.  doi: 10.1016/j.anihpc.2009.06.002.  Google Scholar [3] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000.  Google Scholar [4] R. E. Edwards, Functional Analysis, Dover Publications Inc. New York, 1995.  Google Scholar [5] H. Garcke, J. Kampmann, A. Rätz and M. Röger, A coupled surface-Cahn-Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes, Math. Models Methods Appl. Sci., 26 (2016), 1149-1189.  doi: 10.1142/S0218202516500275.  Google Scholar [6] S. Luckhaus, The Stefan problem with the Gibbs-Thomson law, Preprint Univ. Pisa, 591 (1991). Google Scholar [7] S. Luckhaus and T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differential Equations, 3 (1995), 253-271.  doi: 10.1007/BF01205007.  Google Scholar [8] F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems, volume 135 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2012. An introduction to geometric measure theory. doi: 10.1017/CBO9781139108133.  Google Scholar [9] M. Röger, Solutions for the Stefan problem with Gibbs-Thomson law by a local minimisation, Interfaces Free Bound., 6 (2004), 105-133.  doi: 10.4171/IFB/93.  Google Scholar [10] R. Schätzle, Hypersurfaces with mean curvature given by an ambient {S}obolev function, J. Differential Geom., 58 (2001), 371-420.  doi: 10.4310/jdg/1090348353.  Google Scholar [11] J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65–96. doi: 10.1007/BF01762360.  Google Scholar [12] L. Simon, Lectures on Geometric Measure Theory, volume 3 of Proceedings of the Centre for Mathematical Analysis, Australian National University., Australian National University Centre for Mathematical Analysis, Canberra, 1983.  Google Scholar

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##### References:
 [1] H. Abels and J. Kampmann, On the sharp interface limit of a model for phase separation on biological membranes, Preprint, arXiv: 1811.12489, 2018. Google Scholar [2] H. Abels and M. Röger, Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous, incompressible fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2403-2424.  doi: 10.1016/j.anihpc.2009.06.002.  Google Scholar [3] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000.  Google Scholar [4] R. E. Edwards, Functional Analysis, Dover Publications Inc. New York, 1995.  Google Scholar [5] H. Garcke, J. Kampmann, A. Rätz and M. Röger, A coupled surface-Cahn-Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes, Math. Models Methods Appl. Sci., 26 (2016), 1149-1189.  doi: 10.1142/S0218202516500275.  Google Scholar [6] S. Luckhaus, The Stefan problem with the Gibbs-Thomson law, Preprint Univ. Pisa, 591 (1991). Google Scholar [7] S. Luckhaus and T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differential Equations, 3 (1995), 253-271.  doi: 10.1007/BF01205007.  Google Scholar [8] F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems, volume 135 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2012. An introduction to geometric measure theory. doi: 10.1017/CBO9781139108133.  Google Scholar [9] M. Röger, Solutions for the Stefan problem with Gibbs-Thomson law by a local minimisation, Interfaces Free Bound., 6 (2004), 105-133.  doi: 10.4171/IFB/93.  Google Scholar [10] R. Schätzle, Hypersurfaces with mean curvature given by an ambient {S}obolev function, J. Differential Geom., 58 (2001), 371-420.  doi: 10.4310/jdg/1090348353.  Google Scholar [11] J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65–96. doi: 10.1007/BF01762360.  Google Scholar [12] L. Simon, Lectures on Geometric Measure Theory, volume 3 of Proceedings of the Centre for Mathematical Analysis, Australian National University., Australian National University Centre for Mathematical Analysis, Canberra, 1983.  Google Scholar
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