arXiv subjects
Publications and source records attributed to Giacomo Canevari.
Let $\mathcal{N}$ be a smooth, compact, connected Riemannian manifold without boundary. Let $\mathcal{E}\to\mathcal{N}$ be the Riemannian universal covering of $\mathcal{N}$. For any bounded, smooth domain $Ω\subseteq\mathbb{R}^d$ and any $u\in\mathrm{BV}(Ω, \, \mathcal{N})$, we show that $u$ has a lifting $v\in\mathrm{BV}(Ω, \, \mathcal{E})$. Our result proves a conjecture by Bethuel and Chiron.
We consider minimising $p$-harmonic maps from three-dimensional domains to the real projective plane, for $1<p<2$. These maps arise as least-energy configurations in variational models for nematic liquid crystals. We show that the singular set of such a map decomposes into a $1$-dimensional set, which can be physically interpreted as a non-orientable line defect, and a locally finite set, i.e. a collection of point defects.
We consider a Landau-de Gennes model for a polydisperse, inhomogeneous suspension of colloidal inclusions in a nematic host, in the dilute regime. We study the homogenised limit and compute the effective free energy of the composite material. By suitably choosing the shape of the inclusions and imposing a quadratic, Rapini-Papoular type surface anchoring energy density, we obtain an effective free energy functional with an additional linear term, which may be interpreted as an "effective field" induced by the inclusions. Moreover, we compute the effective free energy in a regime of "very strong anchoring", that is, when the surface energy effects dominate over the volume free energy.
We study nematic equilibria on three-dimensional square wells, with emphasis on Well Order Reconstruction Solutions (WORS) as a function of the well size, characterized by $λ$, and the well height denoted by $ε$. The WORS are distinctive equilibria reported in [10] for square domains, without taking the third dimension into account, which have two mutually perpendicular defect lines running along the square diagonals, intersecting at the square centre. We prove the existence of WORS on three-dimensional wells for arbitrary well heights, with (i) natural boundary conditions and (ii) realistic surface energies on the top and bottom well surfaces, along with Dirichlet conditions on the lateral surfaces. Moreover, the WORS is globally stable for $λ$ small enough in both cases and unstable as $λ$ increases. We numerically compute novel mixed 3D solutions for large $λ$ and $ε$ followed by a numerical investigation of the effects of surface anchoring on the WORS, exemplifying the relevance of the WORS solution in a 3D context.
We consider a Landau-de Gennes model for a suspension of small colloidal inclusions in a nematic host. We impose suitable anchoring conditions at the boundary of the inclusions, and we work in the dilute regime - i.e., the size of the inclusions is much smaller than the typical separation distance between them, so that the total volume occupied by the inclusions is small. By studying the homogenised limit, and proving rigorous convergence results for local minimisers, we compute the effective free energy for the doped material. In particular, we show that not only the phase transition temperature, but any coefficient of the quartic Landau-de Gennes bulk potential can be tuned, by suitably choosing the surface anchoring energy density.
We introduce an operator $\mathbf{S}$ on vector-valued maps $u$ which has the ability to capture the relevant topological information carried by $u$. In particular, this operator is defined on maps that take values in a closed submanifold $N$ of the Euclidean space $\mathbb{R}^m$, and coincides with the distributional Jacobian in case $N$ is a sphere. The range of $\mathbf{S}$ is a set of maps whose values are flat chains with coefficients in a suitable normed abelian group. In this paper, we use $\mathbf{S}$ to characterise strong limits of smooth, $N$-valued maps with respect to Sobolev norms, extending a result by Pakzad and Rivière. We also discuss applications to the study of manifold-valued maps of bounded variation. In a companion paper, we will consider applications to the asymptotic behaviour of minimisers of Ginzburg-Landau type functionals.
We study a modified Landau-de Gennes model for nematic liquid crystals, where the elastic term is assumed to be of subquadratic growth in the gradient. We analyze the behaviour of global minimizers in two- and three-dimensional domains, subject to uniaxial boundary conditions, in the asymptotic regime where the length scale of the defect cores is small compared to the length scale of the domain. We obtain uniform convergence of the minimizers and of their gradients, away from the singularities of the limiting uniaxial map. We also demonstrate the presence of maximally biaxial cores in minimizers on two-dimensional domains, when the temperature is sufficiently low.
We prove the existence of a well order reconstruction solution (WORS) type Landau-de Gennes critical point on a square domain with an isotropic concentric square inclusion, with tangent boundary conditions on the outer square edges. There are two geometrical parameters - the outer square edge length $λ$, and the aspect ratio $ρ$, which is the ratio of the inner and outer square edge lengths. The WORS exists for all geometrical parameters and for all temperatures; we prove that the WORS is globally stable for either $λ$ small enough or for $ρ$ sufficiently close to unity. We study three different types of critical points in this model setting: critical points with the minimal two degrees of freedom consistent with the imposed boundary conditions, critical points with three degrees of freedom and critical points with five degrees of freedom. In the two-dimensional case, we use $Γ$-convergence techniques to identify the energy-minimizing competitors. We decompose the second variation of the Landau-de Gennes energy into three separate components to study the effects of different types of perturbations on the WORS solution and find that it is most susceptible to in-plane perturbations. In the three-dimensional setting, we numerically find up to $28$ critical points for moderately large values of $ρ$ and we find two critical points with the full five degrees of freedom for very small values of $ρ$, with an escaped profile around the isotropic square inclusion.
In this paper we rigorously investigate the emergence of defects on Nematic Shells with genus different from one. This phenomenon is related to a non trivial interplay between the topology of the shell and the alignment of the director field. To this end, we consider a discrete $XY$ system on the shell $M$, described by a tangent vector field with unit norm sitting at the vertices of a triangulation of the shell. Defects emerge when we let the mesh size of the triangulation go to zero, namely in the discrete-to-continuum limit. In this paper we investigate the discrete-to-continuum limit in terms of $Γ$-convergence in two different asymptotic regimes. The first scaling promotes the appearance of a finite number of defects whose charges are in accordance with the topology of shell $M$, via the Poincaré-Hopf Theorem. The second scaling produces the so called Renormalized Energy that governs the equilibrium of the configurations with defects.
We construct an order reconstruction (OR)-type Landau-de Gennes critical point on a square domain of edge length $λ$, motivated by the well order reconstruction solution numerically reported by Kralj and Majumdar. The OR critical point is distinguished by an uniaxial cross with negative scalar order parameter along the square diagonals. The OR critical point is defined in terms of a saddle-type critical point of an associated scalar variational problem. The OR-type critical point is globally stable for small $λ$ and undergoes a supercritical pitchfork bifurcation in the associated scalar variational setting. We consider generalizations of the OR-type critical point to a regular hexagon, accompanied by numerical estimates of stability criteria of such critical points on both a square and a hexagon in terms of material-dependent constants.
We consider the Landau-de Gennes variational model for nematic liquid crystals, in three-dimensional domains. More precisely, we study the asymptotic behaviour of minimizers as the elastic constant tends to zero, under the assumption that minimizers are uniformly bounded and their energy blows up as the logarithm of the elastic constant. We show that there exists a closed set S of finite length, such that minimizers converge to a locally harmonic map away from S. Moreover, S restricted to the interior of the domain is a locally finite union of straight line segments. We provide sufficient conditions, depending on the domain and the boundary data, under which our main results apply. We also discuss some examples.
We study the stability of the radial-hedgehog solution on a three-dimensional (3D) spherical shell with radial boundary conditions, within the Landau-de Gennes theory for nematic liquid crystals. We show that the radial-hedgehog solution has no zeroes for a sufficiently narrow shell, for all temperatures below the nematic supercooling temperature. We prove that the radial-hedgehog solution is the unique global Landau-de Gennes energy minimizer for a sufficiently narrow 3D spherical shell, for all temperatures below the nematic supercooling temperature. We provide explicit geometry-dependent criteria for the global minimality of the radial-hedgehog solution in this temperature regime. In the low temperature limit, we prove the global minimality of the radial-hedgehog solution on a 3D spherical shell, for all values of the inner and outer radii.
In this paper, we study Vanishing Mean Oscillation vector fields on a compact manifold with boundary. Inspired by the work of Brezis and Niremberg, we construct a topological invariant - the index - for such fields, and establish the analogue of Morse's formula. As a consequence, we characterize the set of boundary data which can be extended to nowhere vanishing VMO vector fields. Finally, we show briefly how these ideas can be applied to (unoriented) line fields with VMO regularity, thus providing a reasonable framework for modelling a surface coated with a thin film of nematic liquid crystals.
We consider the Landau-de Gennes variational problem on a bound\-ed, two dimensional domain, subject to Dirichlet smooth boundary conditions. We prove that minimizers are maximally biaxial near the singularities, that is, their biaxiality parameter reaches the maximum value $1$. Moreover, we discuss the convergence of minimizers in the vanishing elastic constant limit. Our asymptotic analysis is performed in a general setting, which recovers the Landau-de Gennes problem as a specific case.
We are concerned with a phase field system consisting of two partial differential equations in terms of the variables thermal displacement, that is basically the time integration of temperature, and phase parameter. The system is a generalization of the well-known Caginalp model for phase transitions, when including a diffusive term for the thermal displacement in the balance equation and when dealing with an arbitrary maximal monotone graph, along with a smooth anti-monotone function, in the phase equation. A Cauchy-Neumann problem has been studied for such a system in arXiv:1107.3950v2 [math.AP], by proving well-posedness and regularity results, as well as convergence of the problem as the coefficient of the diffusive term for the thermal displacement tends to zero. The aim of this contribution is rather to investigate the asymptotic behaviour of the problem as the coefficient in front of the Laplacian of the temperature goes to 0: this analysis is motivated by the types III and II cases in the thermomechanical theory of Green and Naghdi. Under minimal assumptions on the data of the problems, we show a convergence result. Then, with the help of uniform regularity estimates, we discuss the rate of convergence for the difference of the solutions in suitable norms.
We study a diffusion model of phase field type, which consists of a system of two partial differential equations involving as variables the thermal displacement, that is basically the time integration of temperature, and the order parameter. Our analysis covers the case of a non-smooth (maximal monotone) graph along with a smooth anti-monotone function in the phase equation. Thus, the system turns out a generalization of the well-known Caginalp phase field model for phase transitions when including a diffusive term for the thermal displacement in the balance equation. Systems of this kind have been extensively studied by Miranville and Quintanilla. We prove existence and uniqueness of a weak solution to the initial-boundary value problem, as well as various regularity results ensuring that the solution is strong and with bounded components. Then we investigate the asymptotic behaviour of the solutions as the coefficient of the diffusive term for the thermal displacement tends to 0 and prove convergence to the Caginalp phase field system as well as error estimates for the difference of the solutions.