arXiv subjects

Carlos Román

Publications and source records attributed to Carlos Román.

17 recordsLinked to original sources

arXiv2026

First Critical Field in the pinned three-dimensional Ginzburg--Landau Model: A matching upper bound

We continue our study of the first critical field $H_{c_1}$ for extreme type-II superconductors governed by the three-dimensional magnetic Ginzburg--Landau functional with a pinning term $a_\varepsilon$, as introduced in our previous work [arXiv:2507.10915]. Building upon the lower bound for $H_{c_1}$ and the characterization of the Meissner solution, we now establish a matching upper bound for $H_{c_1}$, thereby identifying its leading-order behavior. This result confirms the sharpness of the previously derived lower bound and further elucidates the connection between the onset of vorticity and a weighted variant of the \emph{isoflux problem}. Our argument is prompted by the upper bound construction we developed in [arXiv:2510.14910], based on the Biot--Savart law.

math.AP↗

arXiv2026

Numerical approach to the London Equation of superconductivity

In this work, we propose a general discretization strategy for solving the London equation for type-II superconductors in the whole space $\mathbb{R}^3$. To compute the magnetic field $H_0$, we reformulate the problem for the magnetic potential as a transmission problem and discretize it through a nonstandard FEM-BEM coupling. This formulation accounts for both the bounded interior domain and the unbounded exterior domain without introducing an artificial truncation. We then compute the vector field $B_0$, which arises from the Helmholtz-Hodge decomposition of the magnetic potential in the superconducting sample. This field enters the isoflux problem, which identifies the curves along which vortex nucleation first becomes energetically favorable in the Ginzburg--Landau model of superconductivity. We recast the equations for $B_0$ using the mixed formulation of Kikuchi, in which the divergence-free constraint is imposed weakly, and discretize the resulting problem using a classical $H(\operatorname{curl})$-conforming finite element discretization. We validate our discretization strategy through convergence tests and conclude with an application to the isoflux problem. For a ball under a constant applied magnetic field, the unique maximizer is the diameter aligned with the field. For ellipsoids under a constant applied magnetic field aligned with their major axis, our computations provide numerical evidence of a different behavior in sufficiently elongated, cigar-shaped geometries: off-axis competitors reminiscent of U-shaped vortex configurations attain a larger isoflux ratio than the major axis. Since the major axis is therefore not a maximizer, any off-axis maximizer generates, by rotational symmetry, a continuous family of equivalent configurations, implying non-uniqueness and the presence of a degenerate rotational direction in the isoflux problem.

math.NA↗

arXiv2026

Wrinkling of an elastic sheet floating on a liquid sphere

A thin circular elastic sheet floating on a drop-like liquid substrate is deformed due to incompatibility between the curved substrate and the planar sheet. We adopt a variational viewpoint by minimizing the non-convex membrane energy together with a higher-order convex bending energy. Focusing on thin sheets, we expand the minimum of the energy in terms of a small thickness ratio $h$, and identify the first two terms of this expansion. The leading-order term arises from the minimization of a family of one-dimensional relaxed problems, while for the next-order term we establish lower and upper bounds. This generalizes the previous work [P. Bella and R.V. Kohn. Wrikling of a thin circular sheet bonded to a spherical substrate, Philos. Trans. Roy. Soc. A, 375(2017). arXiv:1611.01781] to the physically relevant case of a liquid substrate.

math.AP↗

arXiv2025

Phase transition thresholds and chiral magnetic fields of general degree

We study a variational problem for the Landau--Lifshitz energy with Dzyaloshinskii--Moriya interactions arising in 2D micromagnetics, focusing on the Bogomol'nyi regime. We first determine the minimal energy for arbitrary topological degree, thereby revealing two types of phase transitions consistent with physical observations. In addition, we prove the uniqueness of the energy minimizer in degrees $0$ and $-1$, and nonexistence of minimizers for all other degrees. Finally, we show that the homogeneous state remains stable even beyond the threshold at which the skyrmion loses stability, and we uncover a new stability transition driven by the Zeeman energy.

math.AP↗

arXiv2025

Novel reaction-diffusion PDE model for fingerprint-like pattern emergence via the Schnakenberg mechanism

Fingerprint analysis and fingerprint identification have been the most widely used tools for human identification. To this day, various models have been proposed to explain how fingerprints are formed, ranging from the fibroblast model, which focuses on cell-collagen interactions, to the buckling of thin layers model, both yielding significant results. In this work, we present a reaction-diffusion model of Schnakenberg type, featuring an anisotropic diffusion matrix that follows the ridge orientations supplied by other traditional fingerprint-generation models, and notably yet allows minutiae -- i.e. characteristic microstructures embedded in fingerprints -- to emerge. The statistical analysis of the minutiae distribution in a randomly generated fingerprint collection is consistent with observations in real fingerprints. The model can numerically generate fingerprint-like patterns corresponding to the four basic classifications -- arches, ulnar loops, radial loops, and whorls -- as well as a variety of derived forms. The generated patterns emerge on a convex domain that mimics the geometry of a fingertip, exhibiting the diverse types of minutiae typically analyzed in fingerprint identification and showing strong agreement with those observed in human fingerprints. This model also provides insight into how levels of certainty in human identification can be achieved when based on minutiae positions. All the algorithms are implemented in an open source software named GenCHSin.

q-bio.QM↗

arXiv2025

Vortex lines interaction in the three-dimensional magnetic Ginzburg--Landau model

We complete our study of the three dimensional Ginzburg--Landau functional with magnetic field, in the asymptotic regime of a small inverse Ginzburg--Landau parameter $\varepsilon$, and near the first critical field $H_{c_1}$ for which the first vortex filaments appear in energy minimizers. Under a nondegeneracy condition, we show a next order asymptotic expansion of $H_{c_1}$ as $\varepsilon \to 0$, and exhibit a sequence of transitions, with vortex lines appearing one by one as the intensity of the applied magnetic field is increased: passing $H_{c_1}$ there is one vortex, then increasing $H_{c_1}$ by an increment of order $\log |\log\varepsilon|$ a second vortex line appears, etc. These vortex lines accumulate near a special curve $Γ_0$, solution to an isoflux problem. We derive a next order energy that the vortex lines must minimize in the asymptotic limit, after a suitable horizontal blow-up around $Γ_0$. This energy is the sum of terms where penalizations of the length of the lines, logarithmic repulsion between the lines and magnetic confinement near $Γ_0$ compete. This elucidates the shape of vortex lines in superconductors.

math.AP↗

arXiv2025

First critical field in the pinned three dimensional Ginzburg-Landau model of superconductivity

We study extreme type-II superconductors described by the three dimensional magnetic Ginzburg-Landau functional incorporating a pinning term $a_\varepsilon(x)$, which we assume to be a bounded measurable function satisfying $b\leq a_\varepsilon(x)\leq 1$ for some constant $b>0$. A hallmark of such materials is the formation of vortex filaments, which emerge when the applied magnetic field exceeds the first critical field $H_{c_1}$. In this work, we provide a lower bound for this critical field and provide a characterization of the Meissner solution, that is, the unique vortexless configuration that globally minimizes the energy below $H_{c_1}$. Moreover, we show that the onset of vorticity is intrinsically linked to a weighted variant of the isoflux problem studied in [C. Román, On the First Critical Field in the Three Dimensional Ginzburg-Landau Model of Superconductivity, Comm. Math. Phys. 367 (2019), no.1, 317--349. arXiv:1802.09390] and [C. Román, É. Sandier and S. Serfaty, Bounded vorticity for the 3D Ginzburg-Landau model and an isoflux problem, Proc. Lond. Math. Soc. (3) 126 (2023), no.3, 1015--1062. arXiv:2110.06858]. A crucial role is played by the $\varepsilon$-level tools developed in [C. Román, Three dimensional vortex approximation construction and $\varepsilon$-level estimates for the Ginzburg-Landau functional, Arch. Ration. Mech. Anal. 231 (2019), no.3, 1531--1614. arXiv:1712.07604], which we adapt to the weighted Ginzburg-Landau framework.

math.AP↗

arXiv2024

On the Meissner state for type-II inhomogeneous superconductors

We consider extreme type-II superconductors modeled by the Ginzburg--Landau energy with a pinning term $a_\varepsilon(x)$, which we assume to be a bounded measurable function such that $b\leq a_\varepsilon(x)\leq 1$ for some constant $b>0$. A crucial feature of this type of superconductors is the occurrence of vortices, which appear above the so-called first critical field $H_{c_1}$. In this paper we estimate this value and characterize the behavior of the Meissner solution, the unique vortexless configuration that globally minimizes the energy below $H_{c_1}$. In addition, we show that beyond this value, for applied fields whose strength is slightly below the so-called superheating field $H_{sh}$, there exists a unique Meissner-type solution that locally minimizes the energy.

math.AP↗

arXiv2024

Play Me Something Icy: Practical Challenges, Explainability and the Semantic Gap in Generative AI Music

This pictorial aims to critically consider the nature of text-to-audio and text-to-music generative tools in the context of explainable AI. As a group of experimental musicians and researchers, we are enthusiastic about the creative potential of these tools and have sought to understand and evaluate them from perspectives of prompt creation, control, usability, understandability, explainability of the AI process, and overall aesthetic effectiveness of the results. One of the challenges we have identified that is not explicitly addressed by these tools is the inherent semantic gap in using text-based tools to describe something as abstract as music. Other gaps include explainability vs. useability, and user control and input vs. the human creative process. The aim of this pictorial is to raise questions for discussion and make a few general suggestions on the kinds of improvements we would like to see in generative AI music tools.

cs.SD↗

arXiv2024

Domain branching in micromagnetism: scaling law for the global and local energies

We study the occurrence of domain branching in a class of $(d+1)$-dimensional sharp interface models featuring the competition between an interfacial energy and a non-local field energy. Our motivation comes from branching in uniaxial ferromagnets corresponding to $d=2$, but our result also covers twinning in shape-memory alloys near an austenite-twinned-martensite interface (corresponding to $d=1$, thereby recovering a result of Conti [Comm. Pure Appl. Math. 53 (2000), 1448-1474. https://doi.org/cgkvm4 ]). We prove that the energy density of a minimising configuration in a large cuboid domain $Q_{L,T}=[-L,L]^d\times [0,T]$ scales like $T^{-\frac{2}{3}}$ (irrespective of the dimension $d$) if $L\gg T^{\frac{2}{3}}$. While this already provides a lot of insight into the nature of minimisers, it does not characterise their behaviour close to the top and bottom boundaries of the sample, i.e. in the region where the branching is concentrated. More significantly, we show that minimisers exhibit a self-similar behaviour near the top and bottom boundaries in a statistical sense through local energy bounds: for any minimiser in $Q_{L,T}$, the energy density in a small cuboid $Q_{\ell,t}$ centred at the top or bottom boundaries of the sample, with side lengths $\ell \gg t^{\frac{2}{3}}$, satisfies the same scaling law, that is, it is of order $t^{-\frac{2}{3}}$.

math.AP↗

arXiv2022

Bounded vorticity for the 3D Ginzburg-Landau model and an isoflux problem

We consider the full three-dimensional Ginzburg-Landau model of superconductivity with applied magnetic field, in the regime where the intensity of the applied field is close to the "first critical field" $H_{c_1}$ at which vortex filaments appear, and in the asymptotics of a small inverse Ginzburg-Landau parameter $\varepsilon$. This onset of vorticity is directly related to an "isoflux problem" on curves (finding a curve that maximizes the ratio of a magnetic flux by its length), whose study was initiated in [Román, C. On the First Critical Field in the Three Dimensional Ginzburg-Landau Model of Superconductivity. Commun. Math. Phys. 367, 317-349 (2019). https://doi.org/10.1007/s00220-019-03306-w] and which we continue here. By assuming a nondegeneracy condition for this isoflux problem, which we show holds at least for instance in the case of a ball, we prove that if the intensity of the applied field remains below $H_{c_1}+ C \log |\log \varepsilon|$, the total vorticity remains bounded independently of $\varepsilon$, with vortex lines concentrating near the maximizer of the isoflux problem, thus extending to the three-dimensional setting a two-dimensional result of [Sandier, E., Serfaty, S. Ginzburg-Landau minimizers near the first critical field have bounded vorticity. Cal Var 17, 17-28 (2003). https://doi.org/10.1007/s00526-002-0158-9]. We finish by showing an improved estimate on the value of $H_{c_1}$ in some specific simple geometries.

math.AP↗

arXiv2021

Unbounded mass radial solutions for the Keller-Segel equation in the disk

We consider the boundary value problem $$ \left\{ \begin{array}{rcll} -Δu+ u -λe^u&=&0,\ u>0 & \mathrm{in}\ B_1(0)\\ \partial_νu&=&0&\mathrm{on}\ \partial B_1(0), \end{array}\right. $$ whose solutions correspond to steady states of the Keller--Segel system for chemotaxis. Here $B_1(0)$ is the unit disk, $ν$ the outer normal to $\partial B_1(0)$, and $λ>0$ is a parameter. We show that, provided $λ$ is sufficiently small, there exists a family of radial solutions $u_λ$ to this system which blow up at the origin and concentrate on $\partial B_1(0)$, as $λ\to 0$. These solutions satisfy $$ \lim_{λ\to 0} \frac{u_λ(0)}{|\lnλ|}=0\quad \mbox{and}\quad 0<\lim_{λ\to 0} \frac{1}{|\lnλ|}\int_{B_1(0)}λe^{u_λ(x)}dx<\infty, $$ having in particular unbounded mass, as $λ\to 0$.

math.AP↗

arXiv2018

On the first critical field in the 3D Ginzburg-Landau model of superconductivity

The Ginzburg-Landau model is a phenomenological description of superconductivity. A crucial feature of type-II superconductors is the occurrence of vortices, which appear above a certain value of the applied magnetic field called the first critical field. In this paper we estimate this value, when the Ginzburg-Landau parameter is large, and we characterize the behavior of the Meissner solution, the unique vortexless configuration that globally minimizes the Ginzburg-Landau energy below the first critical field. In addition, we show that beyond this value, for a certain range of the strength of the applied field, there exists a unique Meissner-type solution that locally minimizes the energy.

math.AP↗

arXiv2017

3D vortex approximation construction and $\varepsilon$-level estimates for the Ginzburg-Landau functional

We provide a quantitative three-dimensional vortex approximation construction for the Ginzburg-Landau functional. This construction gives an approximation of vortex lines coupled to a lower bound for the energy, optimal to leading order, analogous to the 2D ones, and valid for the first time at the $\varepsilon$-level. These tools allow for a new approach to analyze the behavior of global minimizers for the Ginzburg-Landau functional below and near the first critical field in 3D, followed in two forthcoming papers. In addition, they allow to obtain an $\varepsilon$-quantitative product estimate for the study of Ginzburg-Landau dynamics.

math.AP↗

arXiv2017

Large conformal metrics with prescribed scalar curvature

Let $(M,g)$ be an $n-$dimensional compact Riemannian manifold. Let $h$ be a smooth function on $M$ and assume that it has a critical point $ξ\in M$ such that $h(ξ)=0$ and which satisfies a suitable flatness assumption. We are interested in finding conformal metrics $g_λ=u_λ^\frac4{n-2}g$, with $u>0$, whose scalar curvature is the prescribed function $h_λ=λ^2+h$, where $λ$ is a small parameter. In the positive case, i.e. when the scalar curvature $R_g$ is strictly positive, we find a family of bubbling metrics $g_λ$, where $u_λ$ blows-up at the point $ξ$ and approaches zero far from $ξ$ as $λ$ goes to zero. In the general case, if in addition we assume that there exists a non-degenerate conformal metric $g_0=u_0^\frac4{n-2}g$, with $u_0>0$, whose scalar curvature is equal to $h$, then there exists a bounded family of conformal metrics $g_{0,λ}=u_{0,λ}^\frac4{n-2}g$, with $u_{0,λ}>0$, which satisfies $u_{0,λ}\to u_0$ uniformly as $λ\to 0$. Here, we build a second family of bubbling metrics $g_λ$, where $u_λ$ blows-up at the point $ξ$ and approaches $u_0$ far from $ξ$ as $λ$ goes to zero. In particular, this shows that this problem admits more than one solution.

math.AP↗

arXiv2015

Interior bubbling solutions for the critical Lin-Ni-Takagi problem in dimension 3

We consider the problem of finding positive solutions of the problem $Δu - λu +u^5 = 0$ in a bounded, smooth domain $Ω$ in $\mathbb{R}^3$, under zero Neumann boundary conditions. Here $λ$ is a positive number. We analyze the role of Green's function of $-Δ+λ$ in the presence of solutions exhibiting single bubbling behavior at one point of the domain when $λ$ is regarded as a parameter. As a special case of our results, we find and characterize a positive value $λ_*$ such that if $λ-λ^*>0$ is sufficiently small, then this problem is solvable by a solution $u_λ$ which blows-up by bubbling at a certain interior point of $Ω$ as $λ\downarrow λ_*$.

math.AP↗

arXiv2014

Large Conformal metrics with prescribed sign-changing Gauss curvature

Let $(M,g)$ be a two dimensional compact Riemannian manifold of genus $g(M)>1$. Let $f$ be a smooth function on $M$ such that $$f \ge 0, \quad f\not\equiv 0, \quad \min_M f = 0. $$ Let $p_1,\ldots,p_n$ be any set of points at which $f(p_i)=0$ and $D^2f(p_i)$ is non-singular. We prove that for all sufficiently small $λ>0$ there exists a family of "bubbling" conformal metrics $g_λ=e^{u_λ}g$ such that their Gauss curvature is given by the sign-changing function $K_{g_λ}=-f+λ^2$. Moreover, the family $u_λ$ satisfies $$u_λ(p_j) = -4\logλ-2\log \left (\frac 1{\sqrt{2}} \log \frac 1λ\right ) +O(1)$$ and $$λ^2e^{u_λ}\rightharpoonup8π\sum_{i=1}^{n}δ_{p_i},\quad \mbox{as }λ\to 0,$$ where $δ_{p}$ designates Dirac mass at the point $p$.

math.AP↗