Since the pioneering work of Canham and Helfrich, variational formulations involving curvature-dependent functionals, like the classical Willmore functional, have proven useful for shape analysis of biomembranes. We address minimizers of the Canham-Helfrich functional defined over closed surfaces enclosing a fixed volume and having fixed surface area. By restricting attention to axisymmetric surfaces, we prove the existence of global minimizers.
We show that local minimizers of the Canham-Helfrich energy are asymptotically stable with respect to a model for relaxational fluid vesicle dynamics that we already studied in previous papers ([12, 11]). The proof is based on a Lojasiewicz-Simon inequality.
We address the minimization of the Canham-Helfrich functional in presence of multiple phases. The problem is inspired by the modelization of heterogeneous biological membranes, which may feature variable bending rigidities and spontaneous curvatures. With respect to previous contributions, no symmetry of the minimizers is here assumed. Correspondingly, the problem is reformulated and solved in the weaker frame of oriented curvature varifolds. We present a lower semicontinuity result and prove existence of single- and multiphase minimizers under area and enclosed-volume constrains. Additionally…
We consider a Canham-Helfrich-type variational problem defined over closed surfaces enclosing a fixed volume and having fixed surface area. The problem models the shape of multiphase biomembranes. It consists of minimizing the sum of the Canham-Helfrich energy, in which the bending rigidities and spontaneous curvatures are now phase-dependent, and a line tension penalization for the phase interfaces. By restricting attention to axisymmetric surfaces and phase distributions, we extend our previous results for a single phase (arXiv:1202.1979) and prove existence of a global minimizer.
We study the conditions on the physical parameters in the Helfrich bending energy of lipid bilayer vesicles. Among embedded surfaces with a biconcave axisymmetric shape, the variation equation is analyzed in detail. This leads to simple conditions which guarantee the solution the information about the geometry.
Existence and regularity of minimizers for a geometric variational problem is shown. The variational integral models an energy contribution of the interface between two immiscible fluids in the presence of surfactants and includes a Helfrich type contribution, a Frank type contribution and a coupling term between the orientation of the surfactants and the curvature of the interface. Analytical results are proven in a one--dimensional situation for curves.
Let $Ω$ be a smooth bounded axisymmetric set in $\R^3$. In this paper we investigate the existence of minimizers of the so-called neo-Hookean energy among a class of axisymmetric maps. Due to the appearance of a critical exponent in the energy we must face a problem of lack of compactness. Indeed as shown by an example of Conti-De Lellis, a phenomenon of concentration of energy can occur preventing the strong convergence in $W^{1,2}(Ω,\mathbb{R}^3)$ of a minimizing sequence along with the equi-integrability of the cofactors of that sequence. We prove that this phenomenon can only take place on…
On the two dimensional sphere, we consider axisymmetric critical points of an isoperimetric problem perturbed by a long-range interaction term. When the parameter controlling the nonlocal term is sufficiently large, we prove the existence of a local minimizer with arbitrary many interfaces in the axisymmetric class of admissible functions. These local minimizers in this restricted class are shown to be critical points in the broader sense (i.e., with respect to all perturbations). We then explore the rigidity, due to curvature effects, in the criticality condition via several quantitative resu…
In this paper, we are interested in shape optimization problems involving the ge ometry (normal, curvatures) of the surfaces. We consider a class of hypersurface s in $\mathbb{R}^{n}$ satisfying a uniform ball condition and we prove the exist ence of a $C^{1,1}$-regular minimizer for general geometric functionals and cons traints involving the first- and second-order properties of surfaces, such as in $\mathbb{R}^{3}$ problems of the form: $$ \inf \int_{\partial Ω} j_0 [ \mathbf{x},\mathbf{n}(\mathbf{x}) ] dA (\mathbf{x}) + \int_{\partial Ω} j_1 [ \mathbf{x},\mathbf{n}(\mathbf{x}),H(\mathbf{x}…
We consider energy minimizing configurations of a nematic liquid crystal around a spherical colloid particle, in the context of the Landau-de Gennes model. The nematic is assumed to occupy the exterior of a ball of radius r_0, satisfy homeotropic weak anchoring at the surface of the colloid, and approach a uniform uniaxial state at infinity. We study the minimizers in two different limiting regimes: for balls which are small compared to the characteristic length scale r_0<<L, and for large balls, r_0>>L. The relationship between the radius and the anchoring strength W is also relevant. For sma…
We study energy minimization of a continuum Landau-de Gennes energy functional for nematic liquid crystals, in three-dimensional axisymmetric domains and in a restricted class of $\mathbb{S}^1$-equivariant (i.e., axially symmetric) configurations. We assume smooth and nonvanishing $\mathbb{S}^1$-equivariant (e.g. homeotropic) Dirichlet boundary conditions and a physically relevant norm constraint (Lyuksyutov constraint) in the interior. Relying on results in \cite{DMP1} in the nonsymmetric setting, we prove partial regularity of minimizers away from a possible finite set of interior singularit…
We analyze an elastic surface energy which was recently introduced by G. Napoli and L.Vergori to model thin films of nematic liquid crystals. We show how a novel approach that takes into account also the extrinsic properties of the surfaces coated by the liquid crystal leads to considerable differences with respect to the classical intrinsic energy. Our results concern three connected aspects: i) using methods of the calculus of variations, we establish a relation between the existence of minimizers and the topology of the surface; ii) we prove, by a Ginzburg-Landau approximation, the well-pos…
