A Sharp Interface Limit of a Nonlocal Variational Model for Pattern Formation in Biomembranes

Ginster, Janusz; Hayrapetyan, Gurgen; Pešić, Anastasija; Zwicknagl, Barbara

doi:10.1137/23m1559099

SIAM J. Math. Anal.

2024

DOI: 10.1137/23m1559099

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A Sharp Interface Limit of a Nonlocal Variational Model for Pattern Formation in Biomembranes

Janusz Ginster,

Gurgen Hayrapetyan,

Anastasija Pešić

et al.

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Cited by 2 publications

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Variational models for pattern formation in biomembranes—A Γ‐convergence result

Pešić

2023

Proc Appl Math and Mech

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Biological membranes are thin structures that are composed of various components. The different components often form microdomains, called lipid rafts, that are arranged in complex patterns. To explain this pattern formation, variational models based on Cahn–Hilliard‐type energies have been introduced that couple the local composition of the membrane to its local curvature, which renders the resulting functionals nonlocal. In this note, we review and extend recent qualitative results on related variational models. This includes a technique to deal with Neumann‐boundary conditions in the construction of a recovery sequence.

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Variational models for pattern formation in biomembranes—A Γ‐convergence result

Pešić

2023

Proc Appl Math and Mech

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$$\varvec{\Gamma }$$-Convergence and Stochastic Homogenization of Second-Order Singular Perturbation Models for Phase Transitions

Donnarumma

2024

J Nonlinear Sci

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We study the effective behavior of random, heterogeneous, anisotropic, second-order phase transitions energies that arise in the study of pattern formations in physical–chemical systems. Specifically, we study the asymptotic behavior, as $$\varepsilon $$ ε goes to zero, of random heterogeneous anisotropic functionals in which the second-order perturbation competes not only with a double well potential but also with a possibly negative contribution given by the first-order term. We prove that, under suitable growth conditions and under a stationarity assumption, the functionals $$\Gamma $$ Γ -converge almost surely to a surface energy whose density is independent of the space variable. Furthermore, we show that the limit surface density can be described via a suitable cell formula and is deterministic when ergodicity is assumed.

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A Sharp Interface Limit of a Nonlocal Variational Model for Pattern Formation in Biomembranes

Cited by 2 publications

References 39 publications

Variational models for pattern formation in biomembranes—A Γ‐convergence result

Variational models for pattern formation in biomembranes—A Γ‐convergence result

$$\varvec{\Gamma }$$-Convergence and Stochastic Homogenization of Second-Order Singular Perturbation Models for Phase Transitions

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