We prove existence and regularity of minimisers for the Canham-Helfrich energy in the class of weak (possibly branched and bubbled) immersions of the -sphere. This solves (the spherical case) of the minimisation problem proposed by Helfrich in 1973, modelling lipid bilayer membranes. On the way to prove the main res…
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48 results for “Helfrich energy”
The article explores Helfrich flow with spontaneous curvature, finding singularities and convergence behaviors.
problem Understanding the long-time behavior of Helfrich flow with spontaneous curvature.
method Analyzing the gradient flow of a locally area- and volume-constrained Willmore flow, and applying it to the Helfrich flow.
result For negative spontaneous curvature, the Helfrich flow exhibits finite-time singularities; for positive spontaneous curvature, it converges globally.
Lowered regularity assumption for a phase-dependent Helfrich energy equation.
problem Analyzing the phase separation line of the Helfrich energy.
method Used a carefully chosen test function with a signed distance function.
result Regularity assumption lowered from to for the phase separation line.
Stability of Membranesmath.DG
Study the stability of membranes using Helfrich energy and second variation formula.
problem Stability of membranes under various conditions.
method Developed and applied a second variation formula for the Helfrich energy for a class of surfaces.
result Studied the second variation of the area functional for a specific example.
Proves Li-Yau inequality for Helfrich functional, ensuring embeddedness in spherical cases.
problem Ensuring embeddedness of minimizers in the Canham-Helfrich model.
method Proves Li-Yau inequality for Helfrich functional, converting singular volume integral to explicit energy threshold.
result Existence of smoothly embedded minimizers in physically relevant cases.
Discretizes Helfrich-type energies on surfaces using triangular complexes.
problem Discretizing curvature energies on surfaces of specific type.
method Asymptotic lower bound combined with recovery sequence of triangulations and edge director fields.
result Valid discrete versions of integral curvature energies on surfaces.
For every and , we construct a smooth genus surface embedded into the unit ball with area and Willmore energy smaller than . From this we deduce that a minimising sequence for Willmore's energy in the class of genus surfaces embedded in the unit ball with area converges …
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 volu…
Mathematical model describes how red blood cells return to equilibrium.
problem How red blood cells regain equilibrium after deformation.
method Gradient flow of the Canham-Helfrich functional, proving global existence and convergence for spheres and axisymmetric tori.
result Global existence and convergence of smooth solutions for spheres and axisymmetric tori under specific energy conditions.
The Helfrich functional, denoted by H^{c_0}, 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 H^{c_0} among all possible configurations. The functional integrates a sp…
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…
Study on membranes under confinement, proving existence and regularity of minimizers.
problem Existence and regularity of minimizers for constrained Helfrich energy.
method Elliptic system analysis, careful study of measure-valued Lagrange multiplier.
result Optimal regularity for solutions throughout branch points, rigid behavior for unit ball minimizers.
In this paper we study the functional $\SW_{λ_1,λ_2}$, which is the the sum of the Willmore energy, -weighted surface area, and -weighted volume, for surfaces immersed in . This coincides with the Helfrich functional with zero `spontaneous curvature'. Our main result is a complete classification of all …
Flow preserves isoperimetric ratio for immersed surfaces.
problem Preserving isoperimetric ratio in Willmore flow.
method Non-local -gradient flow for Willmore energy.
result Long-time existence and convergence for spherical initial data.
The paper proves a reverse isoperimetric inequality and applies it to analyze surface flows.
problem Analyzing the negative gradient flow of the Willmore energy plus volume.
method Proved a quantitative reverse isoperimetric inequality and applied it to the flow.
result Initial surfaces converge to a round point in finite or infinite time.
Study on surface configurations with curvature and elasticity.
problem Equilibrium configurations of surfaces with curvature and elasticity.
method Investigates the Euler-Helfrich functional, focusing on axially symmetric surfaces and their variational problems.
result Critical surfaces for the Euler-Helfrich functional, if axially symmetric, satisfy a simpler second order variational problem.
Study connects hyperbolic geometry to membrane shapes.
problem Understanding the shapes of biological membranes.
method Relates geometry of hyperbolic space to Helfrich model.
result Establishes a connection between membrane shapes and hyperbolic geometry.
We address the geometric Cauchy problem for surfaces associated to the membrane shape equation describing equilibrium configurations of vesicles formed by lipid bilayers. This is the Euler-Lagrange equation of the Canham-Helfrich-Evans elastic curvature energy subject to constraints on the enclosed volume and the surfa…
New surfaces described that are symmetric and solve a specific equation.
problem Understanding symmetric shapes of membranes.
method Characterized and described axially symmetric Helfrich spheres using the reduced membrane equation.
result These surfaces are symmetric and belong to a specific family.
For a bounded smooth domain in the plane and smooth boundary data we consider the minimisation of the Willmore functional for graphs subject to Dirichlet or Navier boundary conditions. For -regular graphs we show that bounds for the Willmore energy imply area and diameter bounds. We then consider the -lower s…
Recent theoretical advances in elasticity of membranes following Helfrich's spontaneous curvature modelcond-mat.soft
Recent theoretical advances in elasticity of membranes following Helfrich's famous spontaneous curvature model are summarized in this review. The governing equations describing equilibrium configurations of lipid vesicles, lipid membranes with free edges, and chiral lipid membranes are presented. Several analytic solut…
Study finds minimizers for complex membrane models without symmetry assumptions.
problem Minimizing the Canham-Helfrich functional in multiple phases for heterogeneous biological membranes.
method Reformulated as oriented curvature varifolds, proving existence without symmetry assumptions.
result Existence of minimizers for single- and multiphase models under constraints.
In this paper we study the steepest descent -gradient flow of the functional $\SW_{λ_1,λ_2}$, which is the the sum of the Willmore energy, -weighted surface area, and -weighted enclosed volume, for surfaces immersed in . This coincides with the Helfrich functional with zero `spontaneous curvature'.…
Generates tubular and membranous shapes using curvature functionals.
problem Difficult analysis of tubular and membranous shapes.
method Modeling as curvature optimization problem, phase-field formulation, GPU algorithm.
result Wide continuum of shape textures discovered.
Delaunay tori minimize Willmore energy under isoperimetric constraints.
problem Finding minimizers of the Willmore energy under isoperimetric constraints.
method Constructing Delaunay tori using complete elliptic integrals and analyzing their Willmore energy.
result Existence of smoothly embedded tori minimizing the Willmore functional under isoperimetric constraints.
Study on evolving interfaces with complex curvature and density effects.
problem Understanding the dynamics of evolving heterogeneous elastic interfaces.
method Modeling an evolving curve with a density function, analyzing the associated gradient flow evolution.
result Analysis of the preservation and asymptotic behavior of geometric properties in the evolving system.
Study of red blood cells using elastic surface theory.
problem Understanding the shape of red blood cells.
method Used Helfrich-Canham functional to model red blood cells as elastic surfaces.
result Cassinian ovals, except for the round sphere, do not solve the shape equation.
Multicomponent bilayer structures arise as the ubiquitous plasma membrane in cellular biology and as blends of amphiphilic copolymers used in electrolyte membranes, drug delivery, and emulsion stabilization within the context of synthetic chemistry. We develop the multicomponent functionalized Cahn-Hilliard (mFCH) free…
Paper finds relations between Willmore-type energies, weighted areas, and vertical potential energies for cylindrical critical points.
problem Tackles relations between three types of energy functions for cylindrical critical points.
method Uses differential equations and critical point analysis for Willmore-type energies and weighted areas.
result Generating curves coincide for Willmore-type energies and weighted areas, and similar results hold for Willmore-type energies and vertical potential energies.
In this paper we consider the evolution of regular closed elastic curves immersed in . Equipping the ambient Euclidean space with a vector field $\ca:\R^n\rightarrow\R^n$ and a function , we assume the energy of is smallest when the curvature $\k$ of is parallel to $\c = (\ca \cir…
Uniform ball property and existence of optimal shapes for a wide class of geometric functionalsmath.OC
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 satisfying a uniform ball condition and we prove the exist ence of a -regular minimizer for general geometric functionals and c…
Recent years have witnessed a trend that advanced mathematical tools, such as algebraic topology, differential geometry, graph theory, and partial differential equations, have been developed for describing biological macromolecules. These tools have considerably strengthened our ability to understand the molecular mech…
Symmetric hypersurfaces and boundaries in R^n+1 with group actions.
problem Symmetry of hypersurfaces with symmetric boundaries.
method Infinitesimal Lie group actions, Cauchy problem, Morrey's regularity theory, Cauchy-Kovalevskaya Theorem.
result Symmetry inheritance for minimal and CMC hypersurfaces with symmetric boundaries.
Holographic energy equals Hamiltonian energy.
problem Equating holographic and Hamiltonian energies.
method Relative holographic and Hamiltonian energy comparison.
result Holographic energy is identical to Hamiltonian energy.
Optimizes energy efficiency in wireless sensor networks with limited information.
problem Maximizing energy efficiency in energy harvesting wireless sensor networks with limited channel state information.
method Modeling as a Multi-Armed Bandits problem and developing an Upper Confidence Bound algorithm.
result Significant gains in energy efficiency compared to benchmark schemes.
Reduces energy for 4D submanifolds in R^n.
problem Energy reduction for 4D submanifolds in R^n.
method Connected sum energy reduction for fourth-order Willmore energy.
result Established a connected sum energy reduction for the fourth-order Willmore energy.
Let be the energy of some knot for any from certain class of functions. The problem is to find knots with extremal values of energy. We discuss the notion of the locally perturbed knot. The knot circle minimizes some energies and maximizes some others. So, is there any energy such that the circle ne…
New formula connects Loewner energy to moving frames' renormalised energy.
problem Calculating Loewner energy of Jordan curves.
method Using renormalised energy of moving frames.
result Loewner energy as Kähler potential for Weil-Petersson space.
The paper proves -convergence of discrete tangent-point energies to continuous energies and ropelength, with applications to biarc curves.
problem Proving convergence of discrete tangent-point energies to continuous energies and ropelength.
method Using biarc curves and interpolation, the paper proves -convergence of discretized tangent-point energies to the continuous tangent-point energies and ropelength functional.
result Discrete almost minimizing biarc curves converge to ropelength minimizers and minimizers of continuous tangent-point energies.
Quantizes Willmore energy in Riemannian manifolds with bounded energy and area.
problem Quantization of Willmore energy in bounded energy and area conditions.
method Uniform boundedness of Willmore energy and area, weak convergence of maps, and conformal structures in compact domain.
result Quantization of Willmore energy holds under specified conditions.
Energy-efficient DL inference for IoT devices reduces power consumption and improves performance.
problem Energy inefficiency in deep learning models for IoT devices.
method Energy-aware early exiting policy to balance energy consumption and inference accuracy.
result Accuracy and service rate improved up to 25% and 35% respectively.
Discrete geometry model approximates Willmore energy.
problem Approximating the Willmore energy for triangulated surfaces.
method A discrete energy defined in the spirit of discrete differential geometry converges to the Willmore energy.
result The discrete energy converges to the Willmore energy in the sense of -convergence.
Abstract reviews hyperbolic positive energy theorems.
problem Analyzing positive energy theorems for hyperbolic spaces.
method Review of existing literature on asymptotically hyperbolic manifolds.
result Summarizes positive energy theorems for hyperbolic spaces.
Enhanced tabular benchmarks for energy-efficient neural architecture search.
problem Energy consumption in deep learning models.
method Introducing EC-NAS, an enhanced tabular benchmark with energy consumption data.
result EC-NAS reveals a balance between energy usage and accuracy in neural architecture search.
Quantum computing offers energy savings over classical computing.
problem Energy efficiency in computing services.
method Cournot competition model constrained by energy usage.
result Quantum computing firms can outperform classical counterparts in energy efficiency.
Investigates a new four-dimensional energy related to Willmore energy.
problem Exploring a new conformally invariant energy in four dimensions.
method Computed first variation, applied Noether's theorem, investigated other generalizations.
result Critical points of the new energy are smooth and do not include minimal hypersurfaces.
Derives energy-momentum tensor from Standard Model, examines energy conditions.
problem Validating energy conditions in the context of the Standard Model.
method Geometric variational problem on globally hyperbolic manifold, deriving energy-momentum tensor.
result Validates various energy conditions in general relativity.
Study of critical tori for mean curvature energies in Killing submersions.
problem Analyzing surface energies in Killing submersions.
method Symmetry reduction and binormal evolution of critical curves.
result Construction of vertical tori critical for mean curvature energies.