Local solutions to a free boundary problem for the Willmore functional

Alessandroni, Roberta; Kuwert, E.

doi:10.48550/arxiv.1408.6670

Cited by 2 publications

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“…which expresses that the level sets of u ε can only meet ∂Ω at a right angle. This approximates the minimisation problem explored in [AK14]. Another possible boundary condition is…”

Section: Introductionmentioning

confidence: 80%

“…The case that Ω has finite perimeter and ∂ ν u ε = 0 almost everywhere on the reduced boundary is a generalisation of the situation in which ∂Ω ∈ C 2 and the level sets of u ε meet ∂Ω at a ninety degrees angle. Such conditions arise naturally when we search for surfaces of minimal perimeter bounding a prescribed volume and may be useful also for models containing Willmore's energy [AK14].…”

Section: Positive Results On Boundary Regularitymentioning

confidence: 99%

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On the boundary regularity of phase-fields for Willmore's energy

Dondl¹,

Wojtowytsch

2018

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We demonstrate that Radon measures which arise as the limit of the Modica-Mortola measures associated to phase-fields with uniformly bounded diffuse area and Willmore energy may be singular at the boundary of a domain and discuss implications for practical applications. We furthermore give partial regularity results for the phase-fields uε at the boundary in terms of boundary conditions and counterexamples without boundary conditions.

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“…which expresses that the level sets of u ε can only meet ∂Ω at a right angle. This approximates the minimisation problem explored in [AK14]. Another possible boundary condition is…”

Section: Introductionmentioning

confidence: 80%

Section: Positive Results On Boundary Regularitymentioning

confidence: 99%

On the boundary regularity of phase-fields for Willmore's energy

Dondl¹,

Wojtowytsch

2018

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…The Willmore conjecture, proposed by Willmore (1965), was recently resolved by Marques and Neves (2014). Work on the Willmore functional continues to be a very active area, with recent progress made on quantisation (Bernard and Riviere, 2014), the gradient flow Schätzle, 2001, 2002), and boundary value problems (Alessandroni and Kuwert, 2014;Dall'Acqua, 2012;Dall'Acqua et al, 2013;Deckelnick and Grunau, 2009). There are many other works besides those mentioned here -the literature on analysis of the Willmore functional is vast.…”

Section: Introductionmentioning

confidence: 88%

Rigidity and stability of spheres in the Helfrich model

Bernard

Wheeler

2018

Interfaces Free Bound.

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The Helfrich functional, denoted by Hc0, is a mathematical expression proposed by Helfrich (1973) for the natural free energy carried by an elastic phospholipid bilayer. Helfrich theorises that idealised elastic phospholipid bilayers minimise Hc0 among all possible configurations. The functional integrates a spontaneous curvature parameter c0 together with the mean curvature of the bilayer and constraints on area and volume, either through an inclusion of osmotic pressure difference and tensile stress or otherwise. Using the mathematical concept of embedded orientable surface to represent the configuration of the bilayer, one might expect to be able to adapt methods from differential geometry and the calculus of variations to perform a fine analysis of bilayer configurations in terms of the parameters that it depends upon. In this article we focus upon the case of spherical red blood cells with a view to better understanding spherocytes and spherocytosis. We provide a complete classification of spherical solutions in terms of the parameters in the Helfrich model. We additionally present some further analysis on the rigidity and stability of spherocytes. Helfrich (1973) for the natural free energy carried by an elastic phospholipid bilayer. Helfrich theorises that idealised elastic phospholipid bilayers minimise H c 0 among all possible configurations. The functional integrates a spontaneous curvature parameter c 0 together with the mean curvature of the bilayer and constraints on area and volume, either through an inclusion of osmotic pressure difference and tensile stress or otherwise. Using the mathematical concept of embedded orientable surface to represent the configuration of the bilayer, one might expect to be able to adapt methods from differential geometry and the calculus of variations to perform a fine analysis of bilayer configurations in terms of the parameters that it depends upon. In this article we focus upon the case of spherical red blood cells with a view to better understanding spherocytes and spherocytosis. We provide a complete classification of spherical solutions in terms of the parameters in the Helfrich model. We additionally present some further analysis on the rigidity and stability of spherocytes. Disciplines Engineering | Science and Technology Studies

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Local solutions to a free boundary problem for the Willmore functional

Cited by 2 publications

References 0 publications

On the boundary regularity of phase-fields for Willmore's energy

On the boundary regularity of phase-fields for Willmore's energy

Rigidity and stability of spheres in the Helfrich model

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