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Adriano Pisante

Publications and source records attributed to Adriano Pisante.

15 recordsLinked to original sources
arXiv2024

Global weak solutions for the inverse mean curvature flow in the Heisenberg group

We consider the inverse mean curvature flow (IMCF) in the Heisenberg group $(\He^n, d_\varepsilon)$, where $d_\varepsilon$ is distance associated to either $| \cdot |_\varepsilon$, $\varepsilon>0$, the natural family of left-invariant Riemannian metrics, or with their sub-Riemannian counterparts for $\varepsilon=0$. For $\Omega \subseteq \He^n$ an open set with smooth boundary $\Sigma_0=\partial \Omega$ satisfying a uniform exterior gauge-ball condition and bounded complement we show existence of a global weak IMCF of generalized hypersurfaces $\{\Sigma^\varepsilon_s\}_{s \geq 0} \subseteq \mathbb{H}^n$ which are level sets of a proper globally Lipschitz function with logarithmic growth at infinity. Here, both in the Riemannian and in the sub-Riemannian setting, we adopt the weak formulation introduced by Huisken and Ilmanen in \cite{HuiskenIlmanen}, following the approach in \cite{Moser} due to Moser and based on the the link between IMCF and $p$-harmonic functions.

math.AP↗
arXiv2022

Torus-like solutions for the Landau-de Gennes model. Part III: torus vs split minimizers

We study the behaviour of global minimizers of a continuum Landau-de Gennes energy functional for nematic liquid crystals, in three-dimensional axially symmetric domains domains diffeomorphic to a ball (a nematic droplet) and in a restricted class of $\bbS^1$-equivariant configurations. It is known from our previous paper \cite{DMP2} that, assuming smooth and uniaxial (e.g. homeotropic) boundary conditions and a physically relevant norm constraint in the interior (Lyuksyutov constraint), minimizing configurations are either of \emph{torus} or of \emph{split} type. Here, starting from a nematic droplet with the homeotropic boundary condition, we show how singular (split) solutions or smooth (torus) solutions (or even both) for the Euler-Lagrange equations do appear as energy minimizers by suitably deforming either the domain or the boundary data. As a consequence, when minimizers among $\bbS^1$-equivariant configurations are singular we derive symmetry breaking result for the minimization among all competitors.

math.AP↗
arXiv2021

Torus-like solutions for the Landau-de Gennes model. Part II: Topology of $\mathbb{S}^1$-equivariant minimizers

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 singularities lying on the symmetry axis. For a suitable class of domains and boundary data we show that for smooth minimizers (torus solutions) the level sets of the signed biaxiality are generically finite union of tori of revolution. Concerning nonsmooth minimizers (split solutions), we characterize their asymptotic behavior around any singular point in terms of explicit $\mathbb{S}^1$-equivariant harmonic maps into $\mathbb{S}^4$, whence the generic level sets of the signed biaxiality contains invariant topological spheres. Finally, in the model case of a nematic droplet, we provide existence of torus solutions, at least when the boundary data are suitable uniaxial deformations of the radial anchoring, and existence of split solutions for boundary data which are suitable linearly full harmonic spheres.

math.AP↗
arXiv2020

Torus-like solutions for the Landau-de Gennes model. Part I: the Lyuksyutov regime

We study global minimizers of a continuum Landau-De Gennes energy functional for nematic liquid crystals, in three-dimensional domains. Assuming smooth and uniaxial (e.g. homeotropic) boundary conditions and a corresponding physically relevant norm constraint (Lyuksyutov constraint) in the interior, we prove full regularity up to the boundary for the minimizers. As a consequence, in a relevant range (which we call the Lyuksyutov regime) of parameters of the model we show that even without the norm constraint isotropic melting is anyway avoided in the energy minimizing configurations. Finally, we describe a class of boundary data including radial anchoring which yield in both the previous situations as minimizers smooth configurations whose level sets of the biaxiality carry nontrivial topology. Results in this paper will be largely employed and refined in the next papers of our series. In particular, in [DMP2], we will prove that for smooth minimizers in a restricted class of axially symmetric configurations, the level sets of the biaxiality are generically finite union of tori of revolution.

math.AP↗
arXiv2018

On large deviations of interface motions for statistical mechanics models

We discuss the sharp interface limit of the action functional associated to either the Glauber dynamics for Ising systems with Kac potentials or the Glauber+Kawasaki process. The corresponding limiting functionals, for which we provide explicit formulae of the mobility and transport coefficients, describe the large deviations asymptotics with respect to the mean curvature flow.

math-ph↗
arXiv2017

Stochastic Allen-Cahn equation with mobility

We introduce a class of stochastic Allen-Cahn equations with a mobility coefficient and colored noise. For initial data with finite free energy, we analyze the corresponding Cauchy problem on the $d$-dimensional torus in the time interval $[0,T]$. Assuming that $d\le 3$ and that the potential has quartic growth, we prove existence and uniqueness of the solution as a process $u$ in $L^2$ with continuous paths, satisfying almost surely the regularity properties $u\in C([0,T]; H^1)$ and $u\in L^2([0,T];H^2)$.

math.AP↗
arXiv2017

Stochastic Allen-Cahn approximation of the mean curvature flow: large deviations upper bound

Consider the Allen-Cahn equation on the $d$-dimensional torus, $d=2,3$, in the sharp interface limit. As it is well known, the limiting dynamics is described by the motion by mean curvature of the interface between the two stable phases. Here, we analyze a stochastic perturbation of the Allen-Cahn equation and describe its large deviation asymptotics in a joint sharp interface and small noise limit. Relying on previous results on the variational convergence of the action functional, we prove the large deviation upper bound. The corresponding rate function is finite only when there exists a time evolving interface of codimension one between the two stable phases. The zero level set of this rate function is given by the evolution by mean curvature in the sense of Brakke. Finally, the rate function can be written in terms of the sum of two non-negative quantities: the first measures how much the velocity of the interface deviates from its mean curvature, while the second is due to the possible occurrence of nucleation events.

math.PR↗
arXiv2015

Uniaxial versus Biaxial Character of Nematic Equilibria in Three Dimensions

We study global minimizers of the Landau-de Gennes (LdG) energy functional for nematic liquid crystals, on arbitrary three-dimensional simply connected geometries with topologically non-trivial and physically relevant Dirichlet boundary conditions. Our results are specific to the low-temperature limit. We prove (i) that (re-scaled) global LdG minimizers converge uniformly to a (minimizing) limiting harmonic map, away from the singular set of the limiting map; (ii) there exist both a point of maximal biaxiality and a nonempty Lebesgue-null set of uniaxial points near each singular point of the limiting harmonic map (this improves the recent results of \cite{contreraslamy}); (iii) estimates for the size of "strongly biaxial" regions in terms of the reduced temperature $t$. We further show that global LdG minimizers in the restricted class of uniaxial $\Qvec$-tensors cannot be stable critical points of the LdG energy for low temperatures.

math.AP↗
arXiv2014

A Global Compactness type result for Palais-Smale sequences in fractional Sobolev spaces

We extend the Global Compactness result by M. Struwe (Math. Z, 1984) to any fractional Sobolev spaces $\dot{H}^s(Ω)$ for $0<s<N/2$ and $Ω\subset \mathbb{R}^N$ a bounded domain with smooth boundary. The proof is a simple direct consequence of the so-called Profile Decomposition of P. Gerard (ESAIM: Control, Optimisation and Calculus of Variations, 1998).

math.AP↗
arXiv2013

Allen-Cahn Approximation of Mean Curvature Flow in Riemannian manifolds II, Brakke's flows

We prove convergence of solutions to the parabolic Allen-Cahn equation to Brakke's motion by mean curvature in space forms, generalizing previous results from [15] in Euclidean space. We show that a sequence of measures, associated to energy density of solutions of the parabolic Allen-Cahn equation, converges in the limit to a family of rectifiable Radon measures, which evolves by mean curvature flow in the sense of Brakke. A key role is played by a local almost monotonicity formula (a weak counterpart of Huisken's monotonicity formula) proved in [22], to get various density bounds for the limiting measures.

math.AP↗
arXiv2013

Allen-Cahn Approximation of Mean Curvature Flow in Riemannian manifolds I, uniform estimates

We are concerned with solutions to the parabolic Allen-Cahn equation in Riemannian manifolds. For a general class of initial condition we show non positivity of the limiting energy discrepancy. This in turn allows to prove almost monotonicity formula (a weak counterpart of Huisken's monotonicity formula) which gives a local uniform control of the energy densities at small scales. Such results will be used in [40] to extend previous important results from [31] in Euclidean space, showing convergence of solutions to the parabolic Allen-Cahn equations to Brakke's motion by mean curvature in space forms.

math.AP↗
arXiv2013

Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces

We obtain an improved Sobolev inequality in H^s spaces involving Morrey norms. This refinement yields a direct proof of the existence of optimizers and the compactness up to symmetry of optimizing sequences for the usual Sobolev embedding. More generally, it allows to derive an alternative, more transparent proof of the profile decomposition in H^s obtained in [P. Gerard, ESAIM 1998] using the abstract approach of dislocation spaces developed in [K. Tintarev & K. H. Fieseler, Imperial College Press 2007]. We also analyze directly the local defect of compactness of the Sobolev embedding in terms of measures in the spirit of [P. L. Lions, Rev. Mat. Iberoamericana 1985]. As a model application, we study the asymptotic limit of a family of subcritical problems, obtaining concentration results for the corresponding optimizers which are well known when s is an integer ([O. Rey, Manuscripta math. 1989; Z.-C. Han, Ann. Inst. H. Poincare Anal. Non Lineaire 1991], [K. S. Chou & D. Geng, Differential Integral Equations 2000]).

math.AP↗
arXiv2012

Bloch bundles, Marzari-Vanderbilt functional and maximally localized Wannier functions

We consider a periodic Schroedinger operator and the composite Wannier functions corresponding to a relevant family of its Bloch bands, separated by a gap from the rest of the spectrum. We study the associated localization functional introduced by Marzari and Vanderbilt, and we prove some results about the existence and exponential localization of its minimizers, in dimension d < 4. The proof exploits ideas and methods from the theory of harmonic maps between Riemannian manifolds.

math-ph↗
arXiv2010

Phase transitions and minimal hypersurfaces in hyperbolic space]

The purpose of this paper is to investigate the Cahn-Hillard approximation for entire minimal hypersurfaces in the hyperbolic space. Combining comparison principles with minimization and blow-up arguments, we prove existence results for entire local minimizers with prescribed behaviour at infinity. Then, we study the limit as the length scale tends to zero through a $Γ$-convergence analysis, obtaining existence of entire minimal hypersurfaces with prescribed boundary at infinity. In particular, we recover some existence results proved in M. Anderson and U. Lang using geometric measure theory.

math.AP↗