arXiv subjects

Marco Badran

Publications and source records attributed to Marco Badran.

8 recordsLinked to original sources

arXiv2026

Stable solutions in the abelian Higgs model

We classify entire stable solutions with quadratic energy growth in the $4$-dimensional abelian Higgs model. More precisely, we prove that such solutions are holomorphic with respect to some complex structure, namely they satisfy the first order equations introduced by Bradlow, and consequently their nodal sets are holomorphic curves. Conversely, we construct solutions whose nodal set is any prescribed holomorphic curve with quadratic area growth. As a corollary, we show that all stable solutions with linear energy growth in $\mathbb{R}^3$ are two-dimensional Bogomolnyi vortices.

math.AP↗

arXiv2026

A two-dimensional Allen-Cahn theory for interfaces with boundary

We develop a two-dimensional Allen-Cahn theory for interfaces with boundary in the line-bundle framework introduced by Fr\"ohlich and Struwe. The central new object is a model solution on the punctured plane whose nodal set is a half-line. Near the boundary of an interface, this solution plays the role of the heteroclinic profile in the classical interior theory. However, unlike the interior setting, where the analysis in directions normal to the interface reduces to the ODE satisfied by the heteroclinic, the boundary model has non-flat level sets and must be studied as a genuinely two-dimensional elliptic solution. In the first part of this paper, we construct this model solution and develop its stability and invertibility theory. In the second part of this paper, we present the main application: we construct Allen--Cahn sections whose nodal sets concentrate on any prescribed finite collection of disjoint line segments in the plane. The construction uses a Lyapunov-Schmidt reduction, but the boundary introduces a new difficulty: it creates large error terms along the interior of the interface. Controlling these terms requires a refined ansatz and new gluing arguments beyond the standard interior Allen-Cahn theory.

math.AP↗

arXiv2026

Uniform small energy regularity for fractional geometric problems

We prove small energy regularity for a parabolic boundary reaction Ginzburg-Landau problem in the full range $s\in (0,1)$, answering a question posed by Hyder, Segatti, Sire and Wang. We also obtain a similar small energy regularity result for fractional harmonic maps to spheres. Both results are uniform as $s\to 1$.

math.AP↗

arXiv2025

Nonlocal free boundary minimal surfaces

We introduce the nonlocal analogue of the classical free boundary minimal hypersurfaces in an open domain $\Omega$ of $\mathbb{R}^n$ as the (boundaries of) critical points of the fractional perimeter $\operatorname{Per}_s(\cdot,\,\Omega )$ with respect to inner variations leaving $\Omega$ invariant. We deduce the Euler-Lagrange equations and prove a few surprising features, such as the existence of critical points without boundary and a strong volume constraint in $\Omega$ for unbounded hypersurfaces. Moreover, we investigate stickiness properties and regularity across the boundary.

math.AP↗

arXiv2025

Stationary $p$-harmonic maps approaching planar singular harmonic maps to the circle

Given a bounded planar domain $\Omega \subset \mathbb{R}^2$, we show that any singular harmonic map into the circle $\mathbb{S}^1$ corresponding to a topologically nondegenerate critical point of the renormalised energy in the sense of Bethuel, Brezis and H\'elein is a limit of stationary $p$-harmonic maps for $p < 2$ as $p \to 2$

math.AP↗

arXiv2024

Harmonic maps to the circle with higher dimensional singular set

In a closed, oriented ambient manifold $(M^n,g)$ we consider the problem of finding $\mathbb{S}^1$-valued harmonic maps with prescribed singular set. We show that the boundary of any oriented $(n-1)$-submanifold can be realised as the singular set of an $\mathbb{S}^1$-valued map, which is classically harmonic away from the singularity and distributionally harmonic across. If the singular set $\Gamma$ is also embedded and $C^{1,1}$, we consider three variational relaxations of the same problem and show that the energy of minimisers converges, after renormalisation, to the volume $\mathcal{H}^{n-2}(\Gamma)$ plus a lower-order "renormalised energy" -- common to all relaxations -- describing an energetic interaction between different components of the singular set.

math.DG↗

arXiv2022

Solutions of the Ginzburg-Landau equations concentrating on codimension-2 minimal submanifolds

We consider the magnetic Ginzburg-Landau equations in a compact manifold $N$ $$ \begin{cases} -\varepsilon^2 \Delta^{A} u=\frac{1}{2}(1-|u|^2)u,\\ \varepsilon^2 d^*dA=\langle\nabla^A u,iu\rangle \end{cases} $$ formally corresponding to the Euler-Lagrange equations for the energy functional $$ E(u,A)=\frac{1}{2}\int_{N}\varepsilon^2|\nabla^Au|^{2}+\varepsilon^4|dA|^{2}+\frac{1}{4}(1-|u|^{2})^{2}. $$ Here $u:N\to \mathbb{C}$ and $A$ is a 1-form on $N$. Given a codimension-2 minimal submanifold $M\subset N$ which is also oriented and non-degenerate, we construct a solution $(u_\varepsilon,A_\varepsilon)$ such that $u_\varepsilon$ has a zero set consisting of a smooth surface close to $M$. Away from $M$ we have $$ u_\varepsilon(x)\to\frac{z}{|z|},\quad A_\varepsilon(x)\to\frac{1}{|z|^2}(-z^2dz^1+z^1dz^2),\quad x=\exp_y(z^\beta\nu_\beta(y)). $$ as $\varepsilon\to 0$, for all sufficiently small $z\ne 0$ and $y\in M$. Here, $\{\nu_1,\nu_2\}$ is a normal frame for $M$ in $N$. This improves a recent result by De Philippis and Pigati who built a solution for which the concentration phenomenon holds in an energy, measure-theoretical sense.

math.AP↗

arXiv2022

Entire solutions to 4-dimensional Ginzburg-Landau equations and codimension 2 minimal submanifolds

We consider the magnetic Ginzburg-Landau equations in $\mathbb{R}^4$ $$ \begin{cases} -\varepsilon^2(\nabla-iA)^2u = \frac{1}{2}(1-|u|^{2})u,\\ \varepsilon^2 d^*dA = \langle(\nabla-iA)u,iu\rangle \end{cases} $$ formally corresponding to the Euler-Lagrange equations for the energy functional $$ E(u,A)=\frac{1}{2}\int_{\mathbb{R}^4}|(\nabla-iA)u|^{2}+\varepsilon^2|dA|^{2}+\frac{1}{4\varepsilon^2}(1-|u|^{2})^{2}. $$ Here $u:\mathbb{R}^4\to \mathbb{C}$, $A: \mathbb{R}^4\to\mathbb{R}^4$ and $d$ denotes the exterior derivative acting on the one-form dual to $A$. Given a 2-dimensional minimal surface $M$ in $\mathbb{R}^3$ with finite total curvature and non-degenerate, we construct a solution $(u_\varepsilon,A_\varepsilon)$ which has a zero set consisting of a smooth 2-dimensional surface close to $M\times \{0\}\subset \mathbb{R}^4$. Away from the latter surface we have $|u_\varepsilon| \to 1$ and $$ u_\varepsilon(x)\, \to\, \frac {z}{|z|},\quad A_\varepsilon(x)\, \to\, \frac 1{|z|^2} ( -z_2 \nu(y) + z_1 {\textbf{e}}_4), \quad x = y + z_1 \nu(y) + z_2 {\textbf{e}}_4 $$ for all sufficiently small $z\ne 0$. Here $y\in M$ and $\nu(y)$ is a unit normal vector field to $M$ in $\mathbb{R}^3$.

math.AP↗