Local solutions to a free boundary problem for the Willmore functional

“…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.…”

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confidence: 99%

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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“…In such a case we define the second fundamental form of ϕ in local coordinates as II ij (p) = (∂ ij ϕ(p)) ⊥ , Date: November 15, 2019. 1 for any p ∈ Σ \ ∂Σ, where (·) ⊥ denotes the orthogonal projection onto (dϕ(T p Σ)) ⊥ . Denoting by g ij = ∂ i ϕ, ∂ j ϕ the induced metric tensor on Σ and by g ij the components of its inverse, we define the mean curvature vector by H(p) = 1 2 g ij (p)II ij (p), for any p ∈ Σ \ ∂Σ, where sum over repeated indices is understood.…”

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confidence: 99%

“…Elastic surfaces with boundary. If γ = γ 1 ∪ ... ∪ γ α is a finite disjoint union of smooth closed compact embedded curves, a classical formulation of the Plateau's problem with datum γ may be to solve the minimization problem (1) min µ ϕ (Σ) | ϕ : Σ → R 3 , ϕ| ∂Σ : ∂Σ → γ embedding , that is one wants to look for the surface of least area having the given boundary. From a physical point of view, solutions of the Plateau's problem are good models of soap elastic films having the given boundary ( [19]).…”

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confidence: 99%

“…Apart from the above cited [26], Willmore surfaces with a boundary also of the form Γ R,h have been studied together with the rotational symmetry of the surface in [4], [6], [7], [8], and [9]; a new result about symmetry breaking is [16]. Also, interesting results about Willmore surfaces in a free boundary setting is contained in [1]. A relation between Willmore surfaces and minimal surfaces is investigated in [5].…”

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confidence: 99%

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Connected surfaces with boundary minimizing the Willmore energy

Novaga¹,

Pozzetta

2020

Mathematics in Engineering

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For a given family of smooth closed curves γ 1 , ..., γ α ⊂ R 3 we consider the problem of finding an elastic connected compact surface M with boundary γ = γ 1 ∪ ... ∪ γ α . This is realized by minimizing the Willmore energy W on a suitable class of competitors. While the direct minimization of the Area functional may lead to limits that are disconnected, we prove that, if the infimum of the problem is < 4π, there exists a connected compact minimizer of W in the class of integer rectifiable curvature varifolds with the assigned boundary conditions. This is done by proving that varifold convergence of bounded varifolds with boundary with uniformly bounded Willmore energy implies the convergence of their supports in Hausdorff distance. Hence, in the cases in which a small perturbation of the boundary conditions causes the non-existence of Areaminimizing connected surfaces, our minimization process models the existence of optimal elastic connected compact generalized surfaces with such boundary data. We also study the asymptotic regime in which the diameter of the optimal connected surfaces is arbitrarily large. Under suitable boundedness assumptions, we show that rescalings of such surfaces converge to round spheres. The study of both the perturbative and the asymptotic regime is motivated by the remarkable case of elastic surfaces connecting two parallel circles located at any possible distance one from the other. The main tool we use is the monotonicity formula for curvature varifolds ([31], [14]) that we extend to varifolds with boundary, together with its consequences on the structure of varifolds with bounded Willmore energy. Contents 7 4. Perturbative regime: existence in the class of varifolds 10 5. Asymptotic regime: limits of rescalings 14 6. The double circle boundary 16 Appendix A. Curvature varifolds with boundary 18 Appendix B. Monotonicity formula and structure of varifolds with bounded energy 20 References 24 MSC Codes (2010): 49Q20, 74E10, 49J40, 53A05, 49Q10. W(V ) =ˆ| H| 2 dµ V ∈ [0, +∞], if V has generalized mean curvature H, and W(V ) = +∞ otherwise. A rectifiable varifold V = v(M, θ V ) defines a Radon measure on G 2 (R 3 ) := R 3 × G 2,3 , where G 2,3 is the Grassmannian of 2-subspaces of R 3 , identified with the metric space of matrices corresponding to the orthogonal projection on such subspaces. More precisely for any f ∈ C 0 c (G 2 (R 3 )) we define V (f ) :=ˆG 2 (R 3 ) f (p, P ) dV (p, P ) =ˆR 3 f (p, T p M ) dµ V (p).In this way a good notion of convergence in the sense of varifolds is defined, i.e. we say that a sequence V n = v(M n , θ Vn ) converges to V = v(M, θ V ) as varifolds if

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Local solutions to a free boundary problem for the Willmore functional

Cited by 14 publications

References 24 publications

Reflection of Willmore Surfaces with Free Boundaries

Reflection of Willmore Surfaces with Free Boundaries

Rigidity and stability of spheres in the Helfrich model

Connected surfaces with boundary minimizing the Willmore energy

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