arXiv subjects

Giandomenico Orlandi

Publications and source records attributed to Giandomenico Orlandi.

At least 19 recordsLinked to original sources

arXiv2026

Topological point singularities of manifold-valued maps and approximability by smooth maps

Given a positive integer~$p$, we consider $W^{1,p}$-maps from a Euclidean domain of dimension $p+1$ into a closed Riemannian manifold $\NN$. The target manifold is required to satisfy a suitable topological condition, that is, the action of the fundamental group over the~$p$-th homotopy group must be trivial; however, we do \emph{not} assume that $\NN$ is $(p-1)$-connected. Using tools from geometric measure theory --- namely, chains with coefficients in the $p$-th homotopy group of~$\NN$--- we associate to each map a new object that captures its topological point singularities. We provide a characterisation of strong and weak sequential closures of smooth maps in~$W^{1,p}$ in terms of their topological singularities.

math.FA↗

arXiv2026

Well-posedness of nonlinear parabolic equations with unbounded drift via nonlinear evolution theory

We develop a nonlinear evolution framework for nonlinear parabolic equations with unbounded drift terms formulated in Lorentz spaces. The main contribution lies in the construction of uniformly m-accretive operators based on Lorentz-Sobolev embeddings, which allows us to apply the Crandall-Liggett generation theorem for nonlinear evolution equations. Within this framework, we establish existence, uniqueness, and stability of mild solutions. We further show that these mild solutions coincide with weak solutions, ensuring consistency with the variational formulation. Finally, we investigate the long-time asymptotic behavior of solutions.

math.AP↗

arXiv2025

Multiplicity of solutions for Gross-Pitaevskii equations on Riemannian manifolds

We provide a multiplicity result for solutions of time-independent Gross-Pitaevskii equations on closed Riemannian manifolds. Such solutions arise as (possibly non-minimizing) critical points of the Ginzburg-Landau energy having prescribed momentum according to a given tangent velocity field. Lower bounds on the multiplicity of solutions are obtained in terms of the topology of the maximum velocity set, in the small momentum and vorticity core size regime. The proof relies on methods from critical point theory and $Γ$-convergence for Ginzburg-Landau functionals as well as on some new results for codimension 2 isoperimetric-type problems in the small flux regime, possibly of independent interest.

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arXiv2025

$Γ$-convergence of the $p$-Dirichlet energy for manifold-valued maps

We prove a $Γ$-convergence result for the $p$-Dirichlet energy functional defined on maps from a smooth bounded domain $Ω\subseteq \mathbb{R}^{n+k}$ to $\mathscr{N}$, a $(k-2)$-connected and smooth closed Riemannian manifold with Abelian fundamental group, where $n$ and $k$ are integers, $n \geq 0$, $k \geq 2$. We focus on the regime $p \to~k^-$ under Dirichlet boundary conditions. The result provides a description of the asymptotic behavior of the $\textit{topological singular sets}$ for families of $\mathscr{N}$-valued Sobolev maps which satisfy suitable energy bounds. Such topological singular sets are $n$-dimensional flat chains with coefficients in $π_{k-1}(\mathscr{N})$ endowed with a suitable norm. As a consequence of our main result, it follows that the topological singular sets of energy minimizing $p$-harmonic maps converge to a $n$-dimensional flat chain $S$ with coefficients in $π_{k-1}(\mathscr{N})$ which has finite mass and solves the Plateau problem within the homology class associated to the boundary datum.

math.AP↗

arXiv2024

Adapted Lie splitting method for convection-diffusion problems with singular convective term

Splitting methods are a widely used numerical scheme for solving convection-diffusion problems. However, they may lose stability in some situations, particularly when applied to convection-diffusion problems in the presence of an unbounded convective term. In this paper, we propose a new splitting method, called the "Adapted Lie splitting method", which successfully overcomes the observed instability in certain cases. Assuming that the unbounded coefficient belongs to a suitable Lorentz space, we show that the adapted Lie splitting converges to first-order under the analytic semigroup framework. Furthermore, we provide numerical experiments to illustrate our newly proposed splitting approach.

math.NA↗

arXiv2024

Multiplicity results for mass constrained Allen-Cahn equations on Riemannian manifolds with boundary

We present multiplicity results for mass constrained Allen-Cahn equations on a Riemannian manifold with boundary, considering both Neumann and Dirichlet conditions. These results hold under the assumptions of small mass constraint and small diffusion parameter. We obtain lower bounds on the number of solutions according to the Lusternik--Schnirelmann category of the manifold in case of Dirichlet boundary conditions and of its boundary in the case of Neumann boundary conditions. Under generic non-degeneracy assumptions on the solutions, we obtain stronger results based on Morse inequalities. Our approach combines topological and variational methods with tools from Geometric Measure Theory.

math.AP↗

arXiv2023

The Yang-Mills-Higgs functional on complex line bundles: $Γ$-convergence and the London equation

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 $Γ$-convergence result, in the strongly repulsive limit, on the functional rescaled by the logarithm of the coupling parameter. As a corollary, we prove that the energy of minimisers concentrates on an area-minimising surface of dimension $n-2$, while the curvature of minimisers converges to a solution of the London equation.

math.AP↗

arXiv2023

The Yang-Mills-Higgs functional on complex line bundles: asymptotics for critical points

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 parameter tends to zero. After a convenient choice of the gauge, we show compactness of finite-energy critical points in Sobolev norms. Moreover, %independently of the gauge andthanks to a suitable monotonicity formula,we prove that the energy densities of critical points, rescaled by the logarithm of the coupling parameter, concentrate towards the weight measure of a stationary, rectifiable varifold of codimension~2.

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arXiv2023

Energy minimizing maps with prescribed singularities and Gilbert-Steiner optimal networks

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 problem as a homological Plateau problem for rectifiable currents with values in a suitable normed group. This generalizes the pioneering work by Brezis, Coron and Lieb [10].

math.OC↗

arXiv2021

On the obstacle problem for fractional semilinear wave equations

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. moving interfaces) when dealing with singular limits of certain nonlinear wave equations.

math.AP↗

arXiv2021

Topological singular set of vector-valued maps, II: $Γ$-convergence for Ginzburg-Landau type functionals

We prove a $Γ$-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 minimisers, subject to Dirichlet boundary conditions, converges to a generalised surface (more precisely, a flat chain with coefficients in $π_{k-1}(\mathcal{N})$) which solves the Plateau problem in codimension $k$. The analysis relies crucially on the set of topological singularities, that is, the operator $\mathbf{S}$ we introduced in the companion paper arXiv:1712.10203.

math.AP↗

arXiv2020

Anisotropic tempered diffusion equations

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-posedness. We connect the properties of this potential with those of the associated cost function, thus providing a link with optimal transport theory and a supply of new examples of relativistic cost functions. Moreover, we characterize the anisotropic spreading properties of these models and we determine the Rankine-Hugoniot conditions that rule the temporal evolution of jump hypersurfaces under the given anisotropic flows.

math.AP↗

arXiv2019

Lifting for manifold-valued maps of bounded variation

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.

math.FA↗

arXiv2019

Improved partial regularity for manifold-constrained minimisers of subquadratic energies

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.

math.AP↗

arXiv2019

Variational approximation of functionals defined on $1$-dimensional connected sets in $\mathbb{R}^n$

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--Landau type energies proving a $Γ$-convergence result for $n \geq 3$.

math.OC↗

arXiv2019

A variational scheme for hyperbolic obstacle problems

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.

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arXiv2018

Topological singular set of vector-valued maps, I: Applications to manifold-constrained Sobolev and BV spaces

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.

math.AP↗

arXiv2018

Variational approximation of functionals defined on 1-dimensional connected sets: the planar case

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 $Γ$-convergence result. We also introduce a suitable convex relaxation and develop the corresponding numerical implementations. The proposed methods are quite general and the results we obtain can be extended to $n$-dimensional Euclidean space or to more general manifold ambients, as shown in the companion paper [11].

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