arXiv subjects

Brian Seguin

Publications and source records attributed to Brian Seguin.

17 recordsLinked to original sources

arXiv2026

First variation of the fractional $k$-dimensional measure: extending the concept of nonlocal curvature to submanifolds

The fractional $k$-dimensional measure of a submanifold of $\mathbb{R}^n$ is a generalization of the fractional perimeter and fractional length appearing in the literature and depends on a parameter $\sigma$ between $0$ and $1$. Here its first variation is computed. The resulting formula is used to define a nonlocal version of the mean-curvature vector for embedded submanifolds. It is shown that in the case where $k=n-1$, this agrees with the nonlocal mean-curvature that has been widely studied.

math.DG↗

arXiv2025

Construction of an isometric immersion of a bounded, planar region from a framed curve

We develop a framework for characterizing isometric immersions of simply connected, bounded, planar regions with piecewise smooth boundaries into three-dimensional space. Each immersion is associated with a framed curve along the boundary of the image surface, comprised by a parametrized curve and a unit normal vector. We identify a set of compatibility and regularity conditions on this framed curve that ensure the existence of a $C^1$ isometric immersion that is $C^2$ almost everywhere and possesses finite bending energy. Under these conditions, we derive an exact dimensional reduction of the bending energy to a line integral over the boundary curve, without relying on asymptotic assumptions or approximations. By analyzing the behavior of the unit normal vector along the framed boundary, we distinguish between planar and curved regions of the immersed surface. We identify the geometric conditions under which global $C^2$ regularity is potentially lost, in which case the associated immersion belongs to $W^{2,2}$ -- a Sobolev space that arises naturally in variational models of unstretchable elastic surfaces.

math.DG↗

arXiv2025

Approximating the Nonlocal Curvature of Planar Curves

Here we establish several results on the nonlocal curvature of planar curves. First we show how to express the nonlocal curvature of a curve relative to a point in terms of the nonlocal curvatures of simpler components of that curve relative to the same point. To obtain these results, it is necessary to extend the definition of nonlocal curvature to points off of the curve. We also find a formula for the nonlocal curvature of a line segment relative to any point in the plane in terms of the incomplete beta function. These results are then used to prove an approximation theorem, which states that the nonlocal curvature of a planar curve with some H\"older regularity can be approximated by the nonlocal curvature of a linear interpolating spline associated with the curve.

math.DG↗

arXiv2024

A beam that can only bend on the Cantor set

In this work we address the following question: is it possible for a one-dimensional, linearly elastic beam to only bend on the Cantor set and, if so, what would the bending energy of such a beam look like? We answer this question by considering a sequence of beams, indexed by $n$, each one only able to bend on the set associated with the $n$-th step in the construction of the Cantor set and compute the $Γ$-limit of the bending energies. The resulting energy in the limit has a structure similar to the traditional bending energy, a key difference being that the measure used for the integration is the Hausdorff measure of dimension $\ln 2/\ln 3$, which is the dimension of the Cantor set.

math.AP↗

arXiv2023

A definition of fractional k-dimensional measure: bridging the gap between fractional length and fractional area

Here we introduce a fractional notion of $k$-dimensional measure, $0\leq k<n$, that depends on a parameter $σ$ that lies between $0$ and $1$. When $k=n-1$ this coincides with the fractional notions of area and perimeter, and when $k=1$ this coincides with the fractional notion of length. It is shown that, when multiplied by the factor $1-σ$, this $σ$-measure converges to the $k$-dimensional Hausdorff measure up to a multiplicative constant that is computed exactly. We also mention several future directions of research that could be pursued using the fractional measure introduced.

math.CA↗

arXiv2022

On a notion of nonlocal curvature tensor

In the literature various notions of nonlocal curvature can be found. Here we propose a notion of nonlocal curvature tensor. This we do by generalizing an appropriate representation of the classical curvature tensor and by exploiting some analogies with certain fractional differential operators.

math.DG↗

arXiv2018

A fractional notion of length and an associated nonlocal curvature

Here a new notion of fractional length of a smooth curve, which depends on a parameter $\sigma$, is introduced that is analogous to the fractional perimeter functional of sets that has been studied in recent years. It is shown that in an appropriate limit the fractional length converges to the traditional notion of length up to a multiplicative constant. Since a curve that connects two points of minimal length must have zero curvature, the Euler--Lagrange equation associated with the fractional length is used to motivate a nonlocal notion of curvature for a curve. This is analogous to how the fractional perimeter has been used to define a nonlocal mean curvature.

math.DG↗

arXiv2018

A transport theorem for nonconvecting open sets on an embedded manifold

Most transport theorems---that is, a formula for the rate of change of an integral in which both the integrand and domain of integration depend on time---involve domains that evolve according to a flow map. Such domains are said to be convecting. Here a transport theorem for nonconvecting domains evolving on an embedded manifold is established. While the domain is not convecting, it is assumed that the boundary of the domain does evolve according to a flow map is some generalized sense. The proof relies on considering the evolving set as a fixed set in one higher dimension and then using the divergence theorem. The domains considered can be irregular in the sense that their boundaries need only be Lipschitz. Tools from geometric measure theory are used to deal with this irregularity.

math.DG↗

arXiv2017

On the Nonlocal Curvatures of Surfaces with or without Boundary

For surfaces without boundary, nonlocal notions of directional and mean curvatures have been recently given. Here, we develop alternative notions, special cases of which apply to surfaces with boundary. Our main tool is a new fractional or nonlocal area functional for compact surfaces.

math.DG↗

arXiv2016

On the homogenization of a new class of locally periodic microstructures in linear elasticity with residual stress

Many biological and engineering materials have nonperiodic microstructures for which classical periodic homogenization results do not apply. Certain nonperiodic microstructures may be approximated by locally periodic microstructures for which homogenization techniques are available. Motivated by the consideration that such materials are often anisotropic and can posses residual stresses, a broad class of locally periodic microstructures is considered and the resulting effective macroscopic equations are derived. The effective residual stress and effective elasticity tensor are determined by solving unit cell problems at each point in the domain. However, it is found that for a certain class of locally periodic microstructures, solving the unit cell problems at only one point in the domain completely determines the effective elasticity tensor.

math.AP↗

arXiv2015

Homogenization of a system of elastic and reaction-diffusion equations modelling plant cell wall biomechanics

In this paper we present a derivation and multiscale analysis of a mathematical model for plant cell wall biomechanics that takes into account both the microscopic structure of a cell wall coming from the cellulose microfibrils and the chemical reactions between the cell wall's constituents. Particular attention is paid to the role of pectin and the impact of calcium-pectin cross-linking chemistry on the mechanical properties of the cell wall. We prove the existence and uniqueness of the strongly coupled microscopic problem consisting of the equations of linear elasticity and a system of reaction-diffusion and ordinary differential equations. Using homogenization techniques (two-scale convergence and periodic unfolding methods) we derive a macroscopic model for plant cell wall biomechanics.

math.AP↗

arXiv2015

Homogenization of a viscoelastic model for plant cell wall biomechanics

The microscopic structure of a plant cell wall is given by cellulose microfibrils embedded in a cell wall matrix. In this paper we consider a microscopic model for interactions between viscoelastic deformations of a plant cell wall and chemical processes in the cell wall matrix. We consider elastic deformations of the cell wall microfibrils and viscoelastic Kelvin--Voigt type deformations of the cell wall matrix. Using homogenization techniques (two-scale convergence and periodic unfolding methods) we derive macroscopic equations from the microscopic model for cell wall biomechanics consisting of strongly coupled equations of linear viscoelasticity and a system of reaction-diffusion and ordinary differential equations. As is typical for microscopic viscoelastic problems, the macroscopic equations for viscoelastic deformations of plant cell walls contain memory terms. The derivation of the macroscopic problem for degenerate viscoelastic equations is conducted using a perturbation argument.

math.AP↗

arXiv2015

Periodic homogenization and material symmetry in linear elasticity

Here homogenization theory is used to establish a connection between the symmetries of a periodic elastic structure associated with the microscopic properties of an elastic material and the material symmetries of the effective, macroscopic elasticity tensor. Previous results of this type exist but here more general symmetries on the microscale are considered. Using an explicit example, we show that it is possible for a material to be fully anisotropic on the microscale and yet the symmetry group on the macroscale can contain elements other than plus or minus the identity. Another example demon- strates that not all material symmetries of the macroscopic elastic tensor are generated by symmetries of the periodic elastic structure.

math-ph↗

arXiv2015

The impact of microfibril orientations on the biomechanics of plant cell walls and tissues: modelling and simulations

It is known that the orientation of cellulose microfibrils within plant cell walls has an important impact on the morphogenesis of plant cells and tissues. Viewing the shape of a plant cell as a square prism or cylinder with the axis aligning with the primary direction of expansion and growth, the orientation of the microfibrils within the cell wall on the sides of the cell is known. However, not much is known about their orientation at the ends of the cell. Here we investigate the impact of the orientation of cellulose microfibrils within a plant cell wall at the ends of the cell by solving the equations of linear elasticity numerically. Three different scenarios for the orientation of the microfibrils are considered. The macroscopic elastic properties of the cell wall are obtained using homogenization theory from the microscopic description of the elastic properties of the cell wall microfibrils and wall matrix. It is found that the orientation of the microfibrils in the upper and lower parts of cell walls do not affect the expansion of the cell in the direction of its axis but do affect its expansion in the lateral directions. The arrangement of the microfibrils in the upper and lower parts of cell walls is especially important in the case of directed forces acting on plant cell walls and tissues.

q-bio.CB↗

arXiv2014

Calculating the bending moduli of the Canham--Helfrich free-energy density from a particular potential

The Canham--Helfrich free-energy density for a lipid bilayer involves the mean and Gaussian curvatures of the midsurface of the bilayer. The splay and saddle-splay moduli $κ$ and $\barκ$ regulate the sensitivity of the free-energy density to changes of these curvatures. Seguin and Fried derived the Canham--Helfrich energy by taking into account the interactions between the molecules comprising the bilayer, giving rise to integral representations for the moduli in terms of the interaction potential. In the present work, two potentials are chosen and the integrals are evaluated to yield expressions for the moduli, which are found to depend on parameters associated with each potential. These results are compared with values of the moduli found in the current literature.

physics.bio-ph↗

arXiv2013

Stable and unstable helices: Soap films in cylindrical tubes

Cox & Jones recently devised and studied an interesting variant of the classical Plateau problem, a variant in which a helical soap film is confined to a cylindrical tube with circular cross-section. Through experiments, numerics, and some analysis, they found that the length and (inner) radius of the tube strongly influence the equilibrium shape of the confined soap film. In this paper, an area minimization problem associated with determining the shape of the film is formulated and analyzed to determine which surfaces are local minima. The connection between a functional inequality and the associated eigenvalue problem plays an important role in the analysis. For helical films, a more detailed analysis is carried out and stability conditions consistent with the experimental and numerical results of Cox & Jones are obtained.

math.CA↗

arXiv2013

Microphysical derivation of the Canham--Helfrich free-energy density

The Canham--Helfrich free-energy density for a lipid bilayer has drawn considerable attention. Aside from the mean and Gaussian curvatures, this free-energy density involves a spontaneous mean-curvature that encompasses information regarding the preferred, natural shape of the lipid bilayer. We use a straightforward microphysical argument to derive the Canham--Helfrich free-energy density. Our derivation (i) provides a justification for the common assertion that spontaneous curvature originates primarily from asymmetry between the leaflets comprising a bilayer and (ii) furnishes expressions for the splay and saddle-splay moduli in terms of derivatives of the underlying potential.

physics.bio-ph↗