Giandomenico Orlandi

Giandomenico Orlandi
University of Verona | UNIVR · Department of Computer Science

PhD

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86
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1,252
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Publications

Publications (86)
Article
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We consider a gauge-invariant Ginzburg–Landau functional (also known as Abelian Yang–Mills–Higgs model), on Hermitian line bundles over closed Riemannian manifolds of dimension n ≥ 3 {n\geq 3} . Assuming a logarithmic energy bound in the coupling parameter, we study the asymptotic behaviour of critical points in the London limit. After a convenient...
Article
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We consider the Abelian Yang–Mills–Higgs functional, in the non-self dual scaling, on a complex line bundle over a closed Riemannian manifold of dimension n≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\od...
Preprint
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We consider a gauge-invariant Ginzburg-Landau functional (also known as Abelian Yang-Mills-Higgs model) on Hermitian line bundles over closed Riemannian manifolds of dimension $n \geq 3$. Assuming a logarithmic energy bound in the coupling parameter, we study the asymptotic behaviour of critical points in the non-self dual scaling, as the coupling...
Article
Full-text available
We investigate the relation between energy minimizing maps valued into spheres having topological singularities at given points and optimal networks connecting them (e.g., Steiner trees, Gilbert-Steiner irrigation networks). We show the equivalence of the corresponding variational problems, interpreting in particular the branched optimal transport...
Preprint
Full-text available
We consider the Abelian Yang-Mills-Higgs functional, in the non-self dual scaling, on a complex line bundle over a closed Riemannian manifold of dimension $n\geq 3$. This functional is the natural generalisation of the Ginzburg-Landau model for superconductivity to the non-Euclidean setting. We prove a $\Gamma$-convergence result, in the strongly r...
Preprint
We investigate the relation between energy minimizing maps valued into spheres having topological singularities at given points and optimal networks connecting them (e.g. Steiner trees, Gilbert-Steiner irrigation networks). We show the equivalence of the corresponding variational problems, interpreting in particular the branched optimal transport p...
Article
We prove existence of weak solutions to the obstacle problem for semilinear wave equations (including the fractional case) by using a suitable approximating scheme in the spirit of minimizing movements. This extends the results in Bonafini et al. (2019), where the linear case was treated. In addition, we deduce some compactness properties of concen...
Article
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We prove a Γ-convergence result for a class of Ginzburg–Landau type functionals with N-well potentials, where N is a closed and (k-2)-connected submanifold of Rm, in arbitrary dimension. This class includes, for instance, the Landau-de Gennes free energy for nematic liquid crystals. The energy density of minimisers, subject to Dirichlet boundary co...
Preprint
We prove existence of weak solutions to the obstacle problem for semilinear wave equations (including the fractional case) by using a suitable approximating scheme in the spirit of minimizing movements. This extends the results in [9], where the linear case was treated. In addition, we deduce some compactness properties of concentration sets (e.g....
Article
We introduce a functional framework which is specially suited to formulate several classes of anisotropic evolution equations of tempered diffusion type. Under an amenable set of hypothesis involving a very natural potential function, these models can be shown to belong to the entropy solution framework devised by Andreu et al. (2005), therefore en...
Article
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Let N be a smooth, compact, connected Riemannian manifold without boundary. Let E→N be the Riemannian universal covering of N. For any bounded, smooth domain Ω⊆Rd and any u∈BV(Ω,N), we show that u has a lifting v∈BV(Ω,E). Our result proves a conjecture by Bethuel and Chiron.
Preprint
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We prove a $\Gamma$-convergence result for a class of Ginzburg-Landau type functionals with $\mathcal{N}$-well potentials, where $\mathcal{N}$ is a closed and $(k-2)$-connected submanifold of $\mathbb{R}^m$, in arbitrary dimension. This class includes, for instance, the Landau-de Gennes free energy for nematic liquid crystals. The energy density of...
Article
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We consider minimising p-harmonic maps from three-dimensional domains to the real projective plane, for 1<p<2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<2$$\end...
Preprint
We introduce a functional framework which is specially suited to formulate several classes of anisotropic evolution equations of tempered diffusion type. Under an amenable set of hypothesis involving a very natural potential function, these models can be shown to belong to the entropy solution framework devised by 4, 5, therefore ensuring well-pose...
Article
In this paper we consider the Euclidean Steiner tree problem and, more generally, (single sink) Gilbert–Steiner problems as prototypical examples of variational problems involving 1-dimensional connected sets in {\mathbb{R}^{n}} . Following the analysis for the planar case presented in [M. Bonafini, G. Orlandi and E. Oudet, Variational approximatio...
Article
We consider an obstacle problem for (possibly non-local) wave equations, and we prove existence of weak solutions through a convex minimization approach based on a time discrete approximation scheme. We provide the corresponding numerical implementation and raise some open questions.
Preprint
Full-text available
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 $\Omega\subseteq\mathbb{R}^d$ and any $u\in\mathrm{BV}(\Omega, \, \mathcal{N})$, we show that $u$ has a lifting $v\in\mathrm{BV}(\Omega, \, \mat...
Chapter
In this paper we tackle the problem of regularisation for inverse problems in single shell diffusion weighted Diffusion weighted imaging restoration. Our Restoration is to recover a high-resolution and denoised DWI signal, prior to any model fitting. The main contribution of our method is the combination of two regularization terms, one using the i...
Preprint
In this paper we consider the Euclidean Steiner tree problem and, more generally , (single sink) Gilbert-Steiner problems as prototypical examples of variational problems involving 1-dimensional connected sets in $\mathbb{R}^n$. Following the the analysis for the planar case presented in [4], we provide a variational approximation through Ginzburg-...
Preprint
In this paper we consider the Euclidean Steiner tree problem and, more generally , (single sink) Gilbert-Steiner problems as prototypical examples of variational problems involving 1-dimensional connected sets in $\mathbb{R}^n$. Following the the analysis for the planar case presented in [4], we provide a variational approximation through Ginzburg-...
Article
Full-text available
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 sphe...
Preprint
Full-text available
We consider obstacle type problems for (non-local) wave equations, modelled by the (fractional) Laplacian, and we prove existence of weak solutions through a convex minimization approach based on a time discrete approximation scheme. We provide the corresponding numerical implementation and state some open questions related to the problem.
Article
In this short note we announce the main results of [2] about variational problems involving 1-dimensional connected sets in the Euclidean plane, such as for example the Steiner tree problem and the irrigation (Gilbert–Steiner) problem.
Preprint
Full-text available
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-orientab...
Preprint
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 sphe...
Article
Full-text available
We sohw existence and uniqueness results for nonlinear parabolic equations in noncylindrical domains with possible jumps in the time variable
Conference Paper
In this paper we formulate a time-optimal control problem in the space of probability measures endowed with the Wasserstein metric as a natural generalization of the correspondent classical problem in \({\mathbb {R}}^d\) where the controlled dynamics is given by a differential inclusion. The main motivation is to model situations in which we have o...
Article
In this paper we consider variational problems involving 1-dimensional connected sets in the euclidean plane, such as the classical Steiner tree problem and the irrigation (Gilbert-Steiner) problem. We relate them to optimal partition problems and provide a variational approximation through Modica-Mortola type energies proving a full $\Gamma$-conve...
Preprint
In this paper we consider variational problems involving 1-dimensional connected sets in the Euclidean plane, such as the classical Steiner tree problem and the irrigation (Gilbert-Steiner) problem. We relate them to optimal partition problems and provide a variational approximation through Modica-Mortola type energies proving a $\Gamma$-convergenc...
Article
Full-text available
Motivated in part by models arising from mathematical descriptions of Bose-Einstein condensation, we consider total variation minimization problems in which the total variation is weighted by a function that may degenerate near the domain boundary, and the fidelity term contains a weight that may be both degenerate and singular. We develop a genera...
Preprint
We show existence and uniqueness results for nonlinear parabolic equations in noncylindrical domains with possible jumps in the time variable
Article
Full-text available
We show that the parabolic minimal surface equation has an eventual regularization effect, that is, the solution becomes smooth after a (strictly positive) finite time.
Chapter
We survey some recent results on variational and evolution problems concerning a certain class of convex 1-homogeneous functionals for vector-valued maps related to models in phase transitions (Hele-Shaw), superconductivity (Ginzburg-Landau) and superfluidity (Gross-Ktaevskii). Minimizers and gradient flows of such functionals may be characterized...
Article
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We study a class of timelike weakly extremal surfaces in flat Minkowski space $\mathbb R^{1+n}$, characterized by the fact that they admit a $C^1$ parametrization (in general not an immersion) of a specific form. We prove that if the distinguished parametrization is of class $C^k$, then the surface is regularly immersed away from a closed singular...
Article
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Diffusion Magnetic Resonance Imaging (MRI) is a powerful non-invasively method pro- ducing images of biological tissues exploiting the water molecules diffusion into the living tissues under a magnetic field. In the last decade, diffusion MRI data have been widely applied to the study of fiber bundle trajectories into the brain white matter and man...
Article
In this paper we consider the asymptotic behavior of the Ginzburg–Landau model for superconductivity in three dimensions, in various energy regimes. Through an analysis via Γ-convergence, we rigorously derive a reduced model for the vortex density and deduce a curvature equation for the vortex lines. In the companion paper (Baldo et al. Commun. Mat...
Article
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We study some functionals that describe the density of vortex lines in superconductors subject to an applied magnetic field, and in Bose-Einstein condensates subject to rotational forcing, in quite general domains in 3 dimensions. These functionals are derived from more basic models via Gamma-convergence, here and in a companion paper. In our main...
Article
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We propose an approximation scheme for complex-valued functions defined on a smooth domain Ω: the approximating functions have a Ginzburg–Landau energy of the same magnitude as the initial function, but they possess moreover improved bounds on vorticity. As an application, we obtain a variant of a Jacobian estimate first established by Jerrard and...
Article
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We consider the gradient flow of a one-homogeneous functional, whose dual involves the derivative of a constrained scalar function. We show in this case that the gradient flow is related to a weak, generalized formulation of the Hele-Shaw flow. The equivalence follows from a variational representation, which is a variant of well-known variational r...
Conference Paper
Diffusion Magnetic Resonance Imaging (MRI) is used to (non-invasively) study neuronal fibers in the brain white matter. Reconstructing fiber paths from such data (tractography problem) is relevant in particular to study the connectivity between two given cerebral regions. By considering the fiber paths between two given areas as geodesics of a suit...
Article
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We develop a suitable generalization of Almgren's theory of varifolds in a lorentzian setting, focusing on area, first variation, rectifiability, compactness and closure issues. Motivated by the asymptotic behaviour of the scaled hyperbolic Ginzburg-Landau equations, and by the presence of singularities in lorentzian minimal surfaces, we introduce,...
Article
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In this paper we consider the asymptotic behavior of the Ginzburg- Landau model for superconductivity in 3-d, in various energy regimes. We rigorously derive, through an analysis via {\Gamma}-convergence, a reduced model for the vortex density, and we deduce a curvature equation for the vortex lines. In a companion paper, we describe further applic...
Article
Full-text available
Diffusion Magnetic Resonance Imaging (MRI) is a powerful non-invasively method producing images of biological tissues exploiting the water molecules diffusion into the liv-ing tissues under a magnetic field. This technique enhances the highly non–homogenous character of the diffusion medium, revealing underlying microstructure. Recently, this metho...
Article
Full-text available
We study various properties of closed relativistic strings. In particular, we characterize their closure under uniform convergence, extending a previous result by Y. Brenier on graph-like unbounded strings, and we discuss some related examples. Then we study the collapsing profile of uniformly convex planar strings which start with zero initial vel...
Article
We consider the sharp interface limit ε{lunate} → 0+ of the semilinear wave equation □ u + ∇ W (u) / ε{lunate}2 = 0 in R1 + n, where u takes values in Rk, k = 1, 2, and W is a double-well potential if k = 1 and vanishes on the unit circle and is positive elsewhere if k = 2. For fixed ε{lunate} > 0 we find some special solutions, constructed around...
Article
Full-text available
We study various properties of closed relativistic strings. In particular, we characterize their closure under uniform convergence, extending a previous result by Y. Brenier on graph-like unbounded strings, and we discuss some related examples. Then we study the collapsing profile of convex planar strings which start with zero initial velocity, and...
Article
Full-text available
We study various properties of closed relativistic strings. In particular, we characterize their closure under uniform convergence, extending a previous result by Y. Brenier on graph-like unbounded strings, and we discuss some related examples. Then we study the collapsing profile of convex planar strings which start with zero initial velocity, and...
Article
For scalar reaction-diffusion in one space dimension, it is known for a long time that fronts move with an exponentially small speed for potentials with several distinct mini- mizers. The purpose of this paper is to provide a similar result in the case of systems. Our method relies on a careful study of the evolution of localized energy. This appro...
Article
We study the asymptotic behaviour, as ε→0, of a sequence {u ε } of minimizers for the Ginzburg-Landau functional which satisfies local energy bounds of order |logε|. The Jacobians Ju ε are shown to converge, in a suitable sense and up to subsequences, to an area minimizing minimal surface of codimension 2. This is achieved without assumptions on th...
Article
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We have identified the effect of the Wigner-Seitz cell geometry in the strong segregation limit of diblock copolymer melts with strong composition asymmetry. A variational problem is proposed describing the distortions of the chain paths due to the geometric constraints imposed by the cell shape. We computed the geometric excess energies for cylind...
Article
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We consider the sharp interface limit $\epsilon \to 0$ of the semilinear wave equation $u_{tt} - \Delta u + \nabla W(u)/ \epsilon^2 = 0$ in $\mathbf R^{1+n}$, where $u$ takes values in $\mathbf R^k$, $k = 1,2$, and $W$ is a double-well potential if $k = 1$ and vanishes on the unit circle and is positive elsewhere if $k = 2$. For fixed $\epsilon > 0...
Article
The purpose of this note is twofold. Firstly, we survey some of our recent works concerning the dynamics of vortices in the two dimensional parabolic Ginzburg-Landau equation, emphasizing in particular the vortex-phase interaction. Secondly, we wish to supplement the analysis carried out in these work with a discussion on the initial value problem...
Article
We survey some recent work concerning the asymptotic dynamics of vortices in the 2-dimensional parabolic Ginzburg-Landau equation, the interaction of vortices with the phase field and the limiting initial value problem for both vortices and phase. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
Article
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For the two dimensional complex parabolic Ginzburg-Landau equation we prove that, asymptotically, vortices evolve according to a simple ordinary differential equation, which is a gradient flow of the Kirchhoff-Onsager functional. This convergence holds except for a finite number of times, corresponding to vortex collisions and splittings, which we...
Article
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We derive estimates for various quantities which are of interest in the analysis of the Ginzburg-Landau equation, and which we bound in terms of the GL-energy E ε and the parameter ε. These estimates are local in nature, and in particular independent of any boundary condition. Most of them improve and extend earlier results on the subject.
Article
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We study the vortex trajectories for the two-dimensional complex parabolic Ginzburg–Landau equation without a well-preparedness assumption. We prove that the trajectory set is rectifiable, and satisfies a weak motion law. In the case of degree ± 1 vortices, the motion law is satisfied in the classical sense. Moreover, dissipation occurs only at a f...
Article
We present some new results for the asymptotic behavior of the complex parabolic Ginzburg-Landau equation. In particular, we establish that, as the parameter E tends to 0, vorticity evolves according to motion by mean curvature in Brakke's weak formulation. The only assumption we make is a natural energy bound on the initial data. In some cases, we...
Article
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In this article, we describe a natural framework for the vortex dynamics in the complex-valued parabolic Ginzburg-Landau equation in $\R^2$ . This general setting does not rely on any assumption of well-preparedness and has the advantage of being valid even after collision times. We carefully analyze collisions leading to annihilation. A new phenom...
Article
In the first part of this paper we prove that certain functionals of Ginzburg-Landau type for maps from a domain in R n+k into R k converge in a suitable sense to the area func-tional for surfaces of dimension n (Theorem 1.1). In the second part we modify this result in order to include Dirichlet boundary condition (Theorem 5.5), and, as a corollar...
Article
Taking advantage of the “invariance” under conformal transformations of certain elliptic operators and combining it with symmetry results obtained by moving totally geodesic hypersurfaces in Hn, we are able to prove the symmetry of positive solutions of −Δu=f(r,u), in balls in Rn, for a class of nonlinearities that do not satisfy the classical hypo...
Article
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We provide a mathematical proof of the existence of traveling vortex rings solutions to the Gross-Pitaevskii (GP) equation in dimension N greater than or equal to 3. We also extend the asymptotic analysis of the free field Ginzburg-Landau equation to a larger class of equations, including the Ginzburg-Landau equation for superconductivity as well a...
Article
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We discuss a number of results which relate the parabolic Ginzburg-Landau equation with motion by mean curvature. We describe the various concentration phenomena underlying this analysis.
Article
The distributional k-dimensional Jacobian of a map u in the Sobolev space W1,k-1 which takes values in the the sphere Sk-1 can be viewed as the boundary of a rectifiable current of codimension k carried by (part of) the singularity of u which is topologically relevant. The main purpose of this paper is to investigate the range of the Jacobian opera...
Article
We establish a Jacobian estimate in the context of Ginzburg–Landau theory, which was conjectured in a recent work of Bourgain, Brezis and Mironescu. To cite this article: F. Bethuel et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).
Article
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We consider complex-valued solutions u" of the Ginzburg{Landau equation on a smooth bounded simply connected domain of N , N 2, where "> 0 is a small parameter. We assume that the Ginzburg{Landau energy E"(u") veries the bound (natural in the context) E"(u") M0j log "j, where M0 is some given constant. We also make several assumptions on the bounda...
Article
Let Omega be a bounded, simply connected, regular domain of R-n, N greater than or equal to 2. For 0 < epsilon < 1, let u(epsilon): Omega --> C be a smooth solution of the Ginzburg Landau equation in Omega with Dirichlet boundary condition g(epsilon), i.e., [GRAPHICS] We are interested in the asymptotic behavior of u, as c goes to zero under the as...
Article
We consider complex-valued solutions uε of the Ginzburg–Landau on a smooth bounded simply connected domain Ω of , N⩾2 (here ε is a parameter between 0 and 1). We assume that uε=gε on ∂Ω, where |gε|=1 and gε is uniformly bounded in . We also assume that the Ginzburg–Landau energy Eε(uε) is bounded by M0|logε|, where M0 is some given constant. We est...
Article
We derive an approximation of codimension-one integral cycles(and cycles modulo p) in a compact Riemannian manifold bymeans of piecewise regular cycles: we obtain both flat convergence andconvergence of the masses. The theorem is proved by using suitableprincipal bundles with a discrete group. As a byproduct, we give analternative proof of the main...
Article
We generalize to arbitrary dimensions a result obtained by F.H. Lin and T. Rivière [4,5] in dimension three for solutions to the Ginzburg–Landau equation, as well as, in arbitrary dimensions, in the case of minimizing solutions.RésuméNous généralisons en dimension quelconque un résultat de F.H. Lin et T. Rivière [4,5] démontré en dimension trois, p...
Article
Given a compact, oriented Riemannian manifold M, without boundary, and a codimension-one homology class in H* (M, Z) (or, respectively, in H* (M, Zp) with p an odd prime), we consider the problem of finding a cycle of least area in the given class: this is known as the homological Plateau’s problem. We propose an elliptic regularization of this pro...
Article
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We study the Hodge decomposition of L 1-(and measure-) differential forms over a compact manifold without boundary, giving positive results and counterexamples. The theory is then applied to the relaxation and minimization, in cohomology classes, of convex functionals with linear growth. This corresponds to a non-linear version of the Hodge theory...
Article
We define a class ℐp (M,N) of Sobolev maps from a manifoldM into a manifoldN, in such a way that each mapu∈ ℐp (M, N) has a well defined [p]-homotopy type, providedN satisfies a topological hypothesis. Using this, we prove the existence of minimizers in [p]-homotopy classes for some polyconvex variational problems.
Article
Motivated by the works of F. Bethuel, H. Brezis, F. Hélein [5] and of F. Bethuel, T. Rivière [6], an asymptotic analysis is carried out for minimizers of the Ginzburg-Landau functional depending on a parameter ɛ, in the more general case of complex line bundles with prescribed Chern class over compact Riemann surfaces. Such a functional describes a...
Article
Given a smooth, bounded open set Ω⊂Rn and a positive smooth function g:∂Ω→R, we consider the numbers mε=minε2∫Ω|Du|2dx+∫Ω|u|2dx:u∈W(1,2)(Ω),u=g on ∂Ω and we obtain the asymptotic formula mε=εC1(g)+ε2C2(g)+o(ε2), where C1(g)=∫∂Ωg2dHn−1,C2(g)=−12∫∂Ωg2K1dHn−1,K1 being the mean curvature of ∂Ω. This is related to phase transition problems. Asymptotic d...
Article
Giovanni Alberti(*) Summary. - We describe an approach via ¡-convergence to the asymptotic behaviour of (minimizers of) complex Ginzburg-Landau functionals in any space dimension, summa- rizing the results of a joint research with S. Baldo and G. Orlandi (ABO1), (ABO2). Sunto. - Il comportamento asintotico delle soluzioni di certe equazioni ellitti...
Article
We describe an approach via ¡-convergence to the asymptotic behaviour of (minimizers of) complex Ginzburg-Landau functionals in any space dimension, sum- marizing some results of a joint research with S. Baldo and G. Orlandi (3), (4). Summary. - The asymptotic behaviour of solutions of certain variational elliptic equations depending on a parameter...
Article
We wish to give an informal exposition of results and ideas from our paper [Convergence of minimizers with local energy bounds for the Ginzburg-Landau functionals, Indiana Univ. Math. J., preprint at http://www.iumj.indiana.edu/IUMJ/forthcoming.php (2008)]. The aim of our work was to study the asymptotic behavior of a sequence {u ε } of (local) min...

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