A new diffuse-interface approximation of the Willmore flow

Rätz, Andreas; Röger, Matthias

doi:10.1051/cocv/2021013

Cited by 4 publications

(3 citation statements)

References 41 publications

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“…The justification of phase field approximations of geometric evolution laws has a long history. Our analysis closely follows the formal analysis in [25], see also [26,36] and [48]. We formulate here assumptions under which the derivation is rigorous.…”

Section: Convergence Towards the Mean Curvature Flow And Willmore Flo...mentioning

confidence: 93%

“…A number of alternative diffuse approximations of the Willmore energy have been proposed, in particular to enforce the Γ-convergence of approximations for non-smooth limit configurations to the L 1 lower semi-continuous envelope of the Willmore functional [46,47,43,48]. These approximations, however, often lack the simplicity of the standard approximation and its direct relation to applications.…”

Section: Introductionmentioning

confidence: 99%

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"Gradient-free" diffuse approximations of the Willmore functional and Willmore flow

Dabrock,

Knüttel,

Röger

2022

Preprint

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We introduce new diffuse approximations of the Willmore functional and the Willmore flow. They are based on a corresponding approximation of the perimeter that has been studied by Amstutz-van Goethem [Interfaces Free Bound. 14 (2012)]. We identify the candidate for the Γ-convergence, prove the Γ-limsup statement and justify the convergence to the Willmore flow by an asymptotic expansion. Furthermore, we present numerical simulations that are based on the new approximation.

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Section: Convergence Towards the Mean Curvature Flow And Willmore Flo...mentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

"Gradient-free" diffuse approximations of the Willmore functional and Willmore flow

Dabrock,

Knüttel,

Röger

2022

Preprint

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“…For an exhaustive list of references about the approximation of the Willmore functional and other variants of this model we refer to [5] and to the recent paper [15], where the interested reader can also find many numerical simulations.…”

Section: Conjecture Let Nmentioning

confidence: 99%

On a Conjecture of De Giorgi About the Phase-Field Approximation of the Willmore Functional

Bellettini

Freguglia

Picenni

2023

Arch Rational Mech Anal

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In 1991 De Giorgi conjectured that, given $$\lambda >0$$ λ > 0 , if $$\mu _\varepsilon $$ μ ε stands for the density of the Allen-Cahn energy and $$v_\varepsilon $$ v ε represents its first variation, then $$\int [v_\varepsilon ^2 + \lambda ] d\mu _\varepsilon $$ ∫ [ v ε 2 + λ ] d μ ε should $$\Gamma $$ Γ -converge to $$c\lambda {\text {Per}}(E) + k \mathcal {W}(\Sigma )$$ c λ Per ( E ) + k W ( Σ ) for some real constant k, where $${\text {Per}}(E)$$ Per ( E ) is the perimeter of the set E, $$\Sigma =\partial E$$ Σ = ∂ E , $$\mathcal {W}(\Sigma )$$ W ( Σ ) is the Willmore functional, and c is an explicit positive constant. A modified version of this conjecture was proved in space dimensions 2 and 3 by Röger and Schätzle, when the term $$\int v_\varepsilon ^2 \, d\mu _\varepsilon $$ ∫ v ε 2 d μ ε is replaced by $$ \int v_\varepsilon ^2 {\varepsilon }^{-1} dx$$ ∫ v ε 2 ε - 1 d x , with a suitable $$k>0$$ k > 0 . In the present paper we show that, surprisingly, the original De Giorgi conjecture holds with $$k=0$$ k = 0 . Further properties of the limit measures obtained under a uniform control of the approximating energies are also provided.

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Generalized minimizing movements for the varifold Canham–Helfrich flow

Brazda,

Kružík,

Stefanelli

2024

Advances in Calculus of Variations

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The gradient flow of the Canham–Helfrich functional is tackled via the generalized minimizing movements approach. We prove the existence of solutions in Wasserstein spaces of varifolds, as well as upper and lower diameter bounds. In the more regular setting of multiply covered C 1 , 1 {C^{1,1}} surfaces, we provide a Li–Yau-type estimate for the Canham–Helfrich energy and prove the conservation of multiplicity along the evolution.

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A new diffuse-interface approximation of the Willmore flow

Cited by 4 publications

References 41 publications

"Gradient-free" diffuse approximations of the Willmore functional and Willmore flow

"Gradient-free" diffuse approximations of the Willmore functional and Willmore flow

On a Conjecture of De Giorgi About the Phase-Field Approximation of the Willmore Functional

Generalized minimizing movements for the varifold Canham–Helfrich flow

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