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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., Comment: 36 pages

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., Comment: In Proceedings of Explainable AI for the Arts Workshop 2024 (XAIxArts 2024) arXiv:2406.14485

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 $t^{-\frac{2}{3}}$., Comment: 47 pages, 4 figures; version accepted for publication

A thin circular elastic sheet floating on a drop-like liquid substrate gets deformed due to incompatibility between the curved substrate and the planar sheet. We adopt a variational viewpoint by minimizing the non-convex membrane energy plus a higher-order convex bending energy. Being interested in thin sheets, we expand the minimum of the energy in terms of a small thickness $h$, and identify the first two terms of this expansion. The leading order term comes from a minimization of a family of one-dimensional ``relaxed'' problems, while for the next-order term we only identify its scaling law. 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., Comment: 36 pages

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 [Rom\'an, 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., Comment: 50 pages, 4 figures

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., Comment: 32 pages

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., Comment: 73 pages

We consider the boundary value problem $$ \left\{ \begin{array}{rcll} -\Delta u+ u -\lambda e^u&=&0,\ u>0 & \mathrm{in}\ B_1(0)\\ \partial_\nu u&=&0&\mathrm{on}\ \partial B_1(0), \end{array}\right. $$ whose solutions correspond to steady states of the Keller--Segel system for chemotaxis. Here $B_1(0)$ is the unit disk, $\nu$ the outer normal to $\partial B_1(0)$, and $\lambda>0$ is a parameter. We show that, provided $\lambda$ is sufficiently small, there exists a family of radial solutions $u_\lambda$ to this system which blow up at the origin and concentrate on $\partial B_1(0)$, as $\lambda\to 0$. These solutions satisfy $$ \lim_{\lambda\to 0} \frac{u_\lambda(0)}{|\ln\lambda|}=0\quad \mbox{and}\quad 0<\lim_{\lambda\to 0} \frac{1}{|\ln\lambda|}\int_{B_1(0)}\lambda e^{u_\lambda(x)}dx<\infty, $$ having in particular unbounded mass, as $\lambda\to 0$., Comment: 33 pages. This is a major revision of the previous version, which contained a significant error. The final version will appear in Calculus of Variations and Partial Differential Equations

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 $\xi\in M$ such that $h(\xi)=0$ and which satisfies a suitable flatness assumption. We are interested in finding conformal metrics $g_\lambda=u_\lambda^\frac4{n-2}g$, with $u>0$, whose scalar curvature is the prescribed function $h_\lambda=\lambda^2+h$, where $\lambda$ 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 $g_\lambda$, where $u_\lambda$ blows-up at the point $\xi$ and approaches zero far from $\xi$ as $\lambda$ 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 $g_{0,\lambda}=u_{0,\lambda}^\frac4{n-2}g$, with $u_{0,\lambda}>0$, which satisfies $u_{0,\lambda}\to u_0$ uniformly as $\lambda\to 0$. Here, we build a second family of bubbling metrics $g_\lambda$, where $u_\lambda$ blows-up at the point $\xi$ and approaches $u_0$ far from $\xi$ as $\lambda$ goes to zero. In particular, this shows that this problem admits more than one solution., Comment: 30 pages, final version. Accepted for publication in J. Differential Equations

We consider the problem of finding positive solutions of the problem $\Delta u - \lambda u +u^5 = 0$ in a bounded, smooth domain $\Omega$ in $\mathbb{R}^3$, under zero Neumann boundary conditions. Here $\lambda$ is a positive number. We analyze the role of Green's function of $-\Delta +\lambda$ in the presence of solutions exhibiting single bubbling behavior at one point of the domain when $\lambda$ is regarded as a parameter. As a special case of our results, we find and characterize a positive value $\lambda_*$ such that if $\lambda-\lambda^*>0$ is sufficiently small, then this problem is solvable by a solution $u_\lambda$ which blows-up by bubbling at a certain interior point of $\Omega$ as $\lambda \downarrow \lambda_*$., Comment: 26 pages, 1 figure

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 $\lambda>0$ there exists a family of "bubbling" conformal metrics $g_\lambda=e^{u_\lambda}g$ such that their Gauss curvature is given by the sign-changing function $K_{g_\lambda}=-f+\lambda^2$. Moreover, the family $u_\lambda$ satisfies $$u_\lambda(p_j) = -4\log\lambda -2\log \left (\frac 1{\sqrt{2}} \log \frac 1\lambda \right ) +O(1)$$ and $$\lambda^2e^{u_\lambda}\rightharpoonup8\pi\sum_{i=1}^{n}\delta_{p_i},\quad \mbox{as }\lambda \to 0,$$ where $\delta_{p}$ designates Dirac mass at the point $p$., Comment: 29 pages, 1 figure