Antonio Segatti

University of Pavia | UNIPV · Department of Mathematics "F. Casorati"

In this paper we discuss the behavior of the Oseen-Frank model for nematic liquid crystals in the limit of vanishing thickness. More precisely, in a thin slab~$\Omega\times (0,h)$ with~$\Omega\subset \mathbb{R}^2$ and $h>0$ we consider the one-constant approximation of the Oseen-Frank model for nematic liquid crystals. We impose Dirichlet boundary...

We introduce a heat flow associated to half-harmonic maps, which have been introduced by Da Lio and Rivière. Those maps exhibit integrability by compensation in one space dimension and are related to harmonic maps with free boundary. We consider a new flow associated to these harmonic maps with free boundary which is actually motivated by a rather...

We consider the gradient flow of a Ginzburg-Landau functional of the type \begin{document}$ F_ \varepsilon^{ \mathrm{extr}}(u): = \frac{1}{2}\int_M \left| {D u} \right|_g^2 + \left| { \mathscr{S} u} \right|^2_g +\frac{1}{2 \varepsilon^2}\left(\left| {u} \right|^2_g-1\right)^2 \mathrm{vol}_g $\end{document} which is defined for tangent vector fields...

We consider the gradient flow of a Ginzburg-Landau functional of the type \[ F_\varepsilon^{\mathrm{extr}}(u):=\frac{1}{2}\int_M \left|D u\right|_g^2 + \left|\mathscr{S} u\right|^2_g +\frac{1}{2\varepsilon^2}\left(\left|u\right|^2_g-1\right)^2\mathrm{vol}_g \] which is defined for tangent vector fields (here $D$ stands for the covariant derivative)...

We introduce a heat flow associated to half-harmonic maps, which have been introduced by Da Lio and Rivi\`ere. Those maps exhibit integrability by compensation in one space dimension and are related to harmonic maps with free boundary. We consider a new flow associated to these harmonic maps with free boundary which is actually motivated by a rathe...

In this paper we consider the gradient flow of the following Ginzburg-Landau type energy \[ F_\eps(u) := \frac{1}{2}\int_{M}\abs{D u}_g^2 +\frac{1}{2\eps^2}\left(\abs{u}_g^2-1\right)^2\Vg. \] This energy is defined on tangent vector fields on a $2$-dimensional closed and oriented Riemannian manifold $M$ (here $D$ stands for the covariant derivative...

This paper deals with a nonlinear degenerate parabolic equation of order α between 2 and 4 which is a kind of fractional version of the Thin Film Equation. Actually, this one corresponds to the limit value α = 4 while the Porous Medium Equation is the limit α = 2. We prove existence of a nonnegative weak solution for a general class of initial data...

We consider the numerical approximation of the Landau–Lifshitz–Gilbert equation, which describes the dynamics of the magnetization in ferromagnetic materials. In addition to the classical micromagnetic contributions, the energy comprises the Dzyaloshinskii–Moriya interaction, which is the most important ingredient for the enucleation and the stabil...

This paper deals with a nonlinear degenerate parabolic equation of order $\alpha$ between 2 and 4 which is a kind of fractional version of the Thin Film Equation. Actually, this one corresponds to the limit value $\alpha=4$ while the Porous Medium Equation is the limit $\alpha=2$. We prove existence of a nonnegative weak solution for a general clas...

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...

In this note we present some recent results on the Mathematical Analysis of Nematic Shells. The type of results we present deal with the analysis of defectless configurations as well as the analysis of defected configurations. The mathematical tools include Topology, Analysis of Partial Differential Equations as well as Variational Techniques like...

We consider a family of fractional porous media equations, recently studied by Caffarelli and Vazquez. We show the construction of a weak solution as Wasserstein gradient flow of a square fractional Sobolev norm. Energy dissipation inequality, regularizing effect and decay estimates for the L^p norms are established. Moreover, we show that a classi...

This paper develops the so-called Weighted Energy-Dissipation (WED) variational approach for the analysis of gradient flows in metric spaces. This focuses on the minimization of the parameter-dependent global-in-time functional of trajectories \[ \mathcal{I}_\varepsilon[u] = \int_0^{\infty} e^{-t/\varepsilon}\left( \frac12 |u'|^2(t) + \frac1{\varep...

This paper develops the so-called Weighted Energy-Dissipation (WED) variational approach for the analysis of gradient flows in metric spaces. This focuses on the minimization of the parameter-dependent global-in-time functional of trajectories \[ \mathcal{I}_\varepsilon[u] = \int_0^{\infty} e^{-t/\varepsilon}\left( \frac12 |u'|^2(t) + \frac1{\varep...

This paper deals with the Cauchy-Dirichlet problem for the fractional Cahn-Hilliard equation. The main results consist of global (in time) existence of weak solutions, characterization of parabolic smoothing effects (implying under proper condition eventual boundedness of trajectories), and convergence of each solution to a (single) equilibrium. In...

This paper deals with the Cauchy-Dirichlet problem for the fractional Cahn-Hilliard equation. The main results consist of global (in time) existence of weak solutions, characterization of parabolic smoothing effects (implying under proper condition eventual boundedness of trajectories), and convergence of each solution to a (single) equilibrium. In...

We consider the numerical approximation of the Landau-Lifshitz-Gilbert equation, which describes the dynamics of the magnetization in ferromagnetic materials. In addition to the classical micromagnetic contributions, the energy comprises the Dzyaloshinskii-Moriya interaction, which is the most important ingredient for the enucleation and the stabil...

In this paper, we consider a model describing evolution of damage in elastic materials, in which stiffness completely degenerates once the material is fully damaged. The model is written by using a phase transition approach, with respect to the damage parameter. In particular, a source of damage is represented by a quadratic form involving deformat...

We establish new quantitative estimates for localized finite differences of solutions to the Poisson problem for the fractional Laplace operator with homogeneous Dirichlet conditions of solid type settled in bounded domains satisfying the Lipschitz cone regularity condition. We then apply these estimates to obtain (i)~regularity results for solutio...

We consider a family of fractional porous media equations, recently studied by Caffarelli and V\'azquez. We show the construction of a weak solution as Wasserstein gradient flow of a square fractional Sobolev norm. Energy dissipation inequality, regularizing effect and decay estimates for the $L^p$ norms are established. Moreover, we show that a cl...

We establish new quantitative estimates for localized finite differences of solutions to the Poisson problem for the fractional Laplace operator with homogeneous Dirichlet conditions of solid type settled in bounded domains satisfying the Lipschitz cone regularity condition. We then apply these estimates to obtain (i)~regularity results for solutio...

We show non-existence of solutions of the Cauchy problem in $\mathbb{R}^N$ for the nonlinear parabolic equation involving fractional diffusion $\partial_t u + (-\Delta)^s \phi(u)= 0,$ with $0<s<1$ and very singular nonlinearities $\phi$ . More precisely, we prove that when $\phi(u)=-1/u^n$ with $n>0$, or $\phi(u) = \log u$, and we take nonnegative...

In this paper we analyze a nonlinear parabolic equation characterized by a singular diffusion term describing very fast diffusion effects. The equation is settled in a smooth bounded three-dimensional domain and complemented with a general boundary condition of dynamic type. This type of condition prescribes some kind of mass conservation; hence ex...

We introduce a fractional variant of the Cahn-Hilliard equation settled in a bounded domain $\Omega$ of $R^N$ and complemented with homogeneous Dirichlet boundary conditions of solid type (i.e., imposed in the entire complement of $\Omega$). After setting a proper functional framework, we prove existence and uniqueness of weak solutions to the rela...

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...

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 now...

The topology and the geometry of a surface play a fundamental role in determining the equilibrium configurations of thin films of liquid crystals. We propose here a theoretical analysis of a recently introduced surface Frank energy, in the case of two-dimensional nematic liquid crystals coating a toroidal particle. Our aim is to show how a differen...

In this work, we analytically investigate a multi-component system for describing phase separation and damage processes in solids. The model consists of a parabolic diffusion equation of fourth order for the concentration coupled with an elliptic system with material dependent coefficients for the strain tensor and a doubly nonlinear differential i...

A reaction-diffusion problem with an obstacle potential is considered in a bounded domain of $\mathbb{R}^{N}$ . Under the assumption that the obstacle $\mathcal{K}$ is a closed convex and bounded subset of $\mathbb{R}$ " with smooth boundary or it is a closed n-dimensional simplex, we prove that the long-time behavior of the solution semigroup asso...

In this work, we analytically investigate a multi-component system for describing phase separation and damage processes in solids. The model consists of a parabolic diffusion equation of fourth order for the concentration coupled with an elliptic system with material dependent coeffcients for the strain tensor and a doubly nonlinear differential in...

In this paper we derive, starting from the basic principles of Thermodynamics, an extended version of the nonconserved Penrose-Fife phase transition model, in which dynamic boundary conditions are considered in order to take into account interactions with walls. Moreover, we study the well-posedness and the asymptotic behavior of the Cauchy problem...

In this note we report on a new variational principle for Gradient Flows in metric spaces. This new variational formulation consists in a functional defined on entire trajectories whose minimizers converge, in the case in which the energy is geodesically convex, to curves of maximal slope. The key point in the proof is a reformulation of the proble...

In this paper we produce families of complete noncompact Riemannian metrics with positive constant σk-curvature equal to 2−k(nk) by performing the connected sum of a finite number of given n-dimensional Delaunay type solutions, provided 2⩽2k<n2⩽2k<n. The problem is equivalent to solve a second order fully nonlinear elliptic equation.

We present a novel variational approach to gradient-flow evolution in metric spaces. In particular, we advance a functional defined on entire trajectories, whose minimizers converge to curves of maximal slope for geodesically convex energies. The crucial step of the argument is the reformulation of the variational approach in terms of a dynamic pro...

We study a Penrose-Fife phase transition model coupled with homogeneous Neumann boundary conditions. Improving previous results, we show that the initial value problem for this model admits a unique solution under weak conditions on the initial data. Moreover, we prove asymptotic regularization properties of weak solutions.

We develop the long-time analysis for gradient flow equations in metric spaces. In particular, we consider two notions of solutions for metric gradient flows, namely energy and generalized solutions. While the former concept coincides with the notion of curves of maximal slope of Ambrosio et al. (2005) [5], we introduce the latter to include limits...

We consider a hydrodynamic system that models the Smectic-A liquid crystal flow. The model consists of the Navier-Stokes equation for the fluid velocity coupled with a fourth-order equation for the layer variable $\vp$, endowed with periodic boundary conditions. We analyze the long-time behavior of the solutions within the theory of infinite-dimens...

In this paper we produce families of complete non compact Riemannian metrics with positive constant $\sigma_k$-curvature by performing the connected sum of a finite number of given $n$-dimensional Delaunay type solutions, provided $2 \leq 2k < n$. The problem is equivalent to solve a second order fully nonlinear elliptic equation.

We develop the long-time analysis for gradient flow equations in metric spaces. In particular, we consider two notions of solutions for metric gradient flows, namely energy and generalized solutions. While the former concept coincides with the notion of curves of maximal slope, we introduce the latter to include limits of time-incremental approxima...

We study the modified Cahn–Hilliard equation proposed by Galenko etal. in order to account for rapid spinodal decomposition in certain glasses. This equation contains, as additional term, the second-order time derivative of the (relative) concentration multiplied by a (small) positive coefficient e{\varepsilon} . Thus, in absence of viscosity effec...

A reaction-diffusion problem with an obstacle potential is considered in a bounded domain of $\R^N$. Under the assumption that the obstacle $\K$ is a closed convex and bounded subset of $\mathbb{R}^n$ with smooth boundary or it is a closed $n$-dimensional simplex, we prove that the long-time behavior of the solution semigroup associated with this p...

In this paper we are concerned with the uniform attractor for a nonautonomous dynamical system related to the Frémond thermo-mechanical model of shape memory alloys. The dynamical system consists of a diffusive equation for the phase proportions coupled with the hyperbolic momentum balance equation, in the case when a damping term is considered in...

Modelling a crystal with impurities we study an atomic chain of point masses with linear nearest neighbour interactions. We assume that the masses of the particles are normalised to 1, except for one heavy particle which has mass M . We investigate the macroscopic behaviour of such a system when M is large, and time and space are scaled accordingly...

This paper addresses the long-time behaviour of gradient flows of nonconvex functionals in Hilbert spaces. Exploiting the notion of generalized semiflows by J. M. Ball, we provide some sufficient conditions for the existence of a global attractor. The abstract results are applied to various classes of nonconvex evolution problems. In particular, we...

In this paper we consider the hyperbolic relaxation of the Cahn-Hilliard equation ruling the evolution of the relative concentration u of one component of a binary alloy system located in a bounded and regular domain of R3. This equation is characterized by the presence of the additional inertial term "utt that accounts for the relaxation of the di...

In this note we summarize some results of a forthcoming paper (see [15]), where we examine, in particular, the long time behavior of the so-called quasistationary phase field model by using a gradient flow approach. Our strategy in fact, is inspired by recent existence results which show that gradient flows of suitable non-convex functionals yield...

A doubly nonlinear parabolic equation of the form $\alpha(u_t)-\Delta u+W'(u)= f$, complemented with initial and either Dirichlet or Neumann homogeneous boundary conditions, is addressed. The two nonlinearities are given by the maximal monotone function $\alpha$ and by the derivative $W'$ of a smooth but possibly nonconvex potential $W$; $f$ is a k...

This paper addresses a doubly nonlinear parabolic inclusion of the form $A(u_t)+B(u)\ni f$. Existence of a solution is proved under suitable monotonicity, coercivity, and structure assumptions on the operators $A$ and $B$, which in particular are both supposed to be subdifferentials of functionals on $L^2(\Omega)$. Moreover, under additional hypoth...

In this paper we consider the Cauchy problem for the abstract nonlinear evolution equation in a Hilbert space ℋ { A(u′(t)) + B(u(t)) - λu(t) ∋ f in ℋ for a.e. t ∈ (0, +∞) u(0) = u 0, where A is a maximal (possibly multivalued) monotone operator from the Hilbert space ℋ to itself, while B is the subdifferential of a proper, convex and lower semicont...

We consider a model describing the evolution of damage in visco-elastic materials, where both the stiffness and the viscosity properties are assumed to degenerate as the damaging is complete. The equation of motion ruling the evolution of macroscopic displacement is hyperbolic. The evolution of the damage parameter is described by a doubly nonlinea...

This paper deals with a fully implicit time discretization scheme with variable time-step for a nonlinear system modelling phase transition and mechanical deformations in shape memory alloys. The model is studied in the non-stationary case and accounts for local microscopic interactions between the phases introducing the gradients of the phase para...

This work deals with a nonlinear system modelling solid-solid phase transitions and mechanical deformations in shape memory alloys. The model is studied in the non-stationary case and accounts for local microscopic interactions between the different phases introducing the gradients of the phase parameters as state variables. By using an approximati...