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Francesca Da Lio

Publications and source records attributed to Francesca Da Lio.

At least 19 recordsLinked to original sources
arXiv2026

Improved Morse Index Stability for Sequences of Harmonic Maps from Degenerating Riemann Surfaces

We study the stability of the extended Morse index, defined as the number of negative and zero eigenvalues of the Jacobi operator, for sequences of harmonic maps on degenerating Riemann surfaces. As the conformal structure approaches the boundary of moduli space, collar collapse creates major analytical challenges. We analyze the second variation of the energy under these degenerations and identify conditions ensuring upper semicontinuity of the extended Morse index. Refining earlier results of the first and second authors in [7], we obtain sharper control of the spectrum of the Jacobi operator on degenerating domains. A key new aspect is the explicit contribution of geodesics arising as limits of the images of degenerating collars. We show that these neck regions converge to geodesic segments whose Morse index contributes nontrivially to the limiting extended index.

math.DG↗
arXiv2025

Morse Index Stability for Sequences of Sacks-Uhlenbeck Maps into a Sphere

In this paper we consider sequences of $p$-harmonic maps, $p>2$, from a closed Riemann surface $Σ$ into the $n$-dimensional sphere $\mathbb{S}^n$ with uniform bounded energy. These are critical points of the energy $E_p(u) :=\int_Σ\left( 1+|{\nabla u}|^2\right)^{p/2} \ dvol_Σ.$ Our two main results are an improved pointwise estimate of the gradient in the neck regions around blow up points and the proof that the necks are asymptotically not contributing to the negativity of the second variation of the energy $E_p.$ This allows us, in the spirit of the paper of the first and second authors in collaboration with M. Gianocca {\em Morse index stability for critical points to conformally invariant Lagrangians}, to show the upper semicontinuity of the Morse index plus nullity for sequences of $p$-harmonic maps into a sphere.

math.AP↗
arXiv2024

Morse Index Stability for the Ginzburg-Landau Approximation

In this paper we study the behaviour of critical points of the Ginzburg-Landau perturbation of the Dirichlet energy into the sphere $E_\varepsilon(u):=\int_Σ\frac{1}{2}|du|^2_h\ \,dvol_h +\frac{1}{4\varepsilon^2}(1-|u|^2)^2\,dvol_h=\int_Σe_{\varepsilon}(u)$. Our first main result is a precise point-wise estimate for $e_\varepsilon(u_k)$ in the regions where compactness fails, which also implies the $L^{2,1}$ quantization in the bubbling process. Our second main result consists in applying the method developed in a previous joint paper with T. Rivière to study the upper-semi-continuity of the extended Morse index to sequences of critical points of $E_ε$: given a sequence of critical points $u_{\varepsilon_k}:Σ\to \mathbb{R}^{n+1}$ of $E_\varepsilon$ that converges in the bubble tree sense to a harmonic map $u_\infty\in W^{1,2}(Σ,{S}^{n})$ and bubbles $v^i_{\infty}:\mathbb{R}^2\to {S}^{n}$, we show that the extended Morse indices of the maps $v^i,u_\infty$ control the extended Morse index of the sequence $u_{\varepsilon_k}$ for $k$ large enough.

math.DG↗
arXiv2023

Blow-up Analysis of Stationary Solutions to a Liouville-Type Equation in 3-D

In this paper we study the asymptotic behavior of sequences of stationary weak solutions to the following Liouville-type equation $-Δu=e^u~~~{in }~~~Ω$, where $Ω$ is an open set of $R^3$. By improving the partial regularity estimates obtained by the first author for the above equation, we succeed in performing a blow-up analysis without Morrey-type assumptions on the solutions $u$ and on the nonlinearity $e^u.$

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arXiv2023

Morse Index Stability for Critical Points to Conformally invariant Lagrangians

We prove the upper-semi-continuity of the Morse index plus nullity of critical points to general conformally invariant Lagrangians in dimension 2 under weak convergence. Precisely we establish that the sum of the Morse indices and the nullity of an arbitrary sequence of weakly converging critical points to a general conformally invariant Lagrangians of maps from an arbitrary closed surface into an arbitrary closed smooth manifold passes to the limit in the following sense : it is asymptotically bounded from above by the sum of the Morse indices plus the nullity of the weak limit and the bubbles, while it was well known that the sum of the Morse index of the weak limit with the Morse indices of the bubbles is asymptotically bounded from above by the Morse indices of the weakly converging sequence. The main result is then extended to the case of sequences of maps from sequences of domains degenerating to a punctured Riemann surface assuming that the lengths of the images by the maps of the collars associated to this degeneration stay below some critical length.

math.DG↗
arXiv2021

A fractional version of Rivière's GL(N)-gauge

We prove that for antisymmetric vectorfield $Ω$ with small $L^2$-norm there exists a gauge $A \in L^\infty \cap \dot{W}^{1/2,2}(\mathbb{R}^1,GL(N))$ such that ${\rm div}_{\frac12} (AΩ- d_{\frac{1}{2}} A) = 0$. This extends a celebrated theorem by Rivière to the nonlocal case and provides conservation laws for a class of nonlocal equations with antisymmetric potentials, as well as stability under weak convergence.

math.AP↗
arXiv2021

Integrability by compensation for Dirac Equation

We consider the Dirac Operator acting on the Clifford Algebra ${C\ell}_{m}$. We show that under critical assumptions on the potential and the spinor field the equation is subject to an integrability by compensation phenomenon and has a sub-critical behaviour below some positive energy threshold (i.e. $ε-$regularity theorem). This extends in 4 space dimensions as well as in 3 dimensions a similar result obtained previously by the two first authors in 2 D in \cite{DLR1}.

math.AP↗
arXiv2020

Bergman-Bourgain-Brezis-type Inequality

In this note, we prove a fractional version in $1$-D of the Bourgain-Brezis inequality \cite{bourgain1}. We show that such an inequality is equivalent to the fact that a holomorphic function $f\colon\D\to\C$ belongs to the Bergman space ${\mathcal{A}}^2(\D)$, namely $f\in L^2(\D)$, if and only if $$\|f\|_{ L^1+ {H}^{-1/2}(S^1)}:=\limsup_{r\to 1^-}\|f(re^{iθ})\|_{ L^1+ {H}^{-1/2}(S^1)}<+\infty.$$ Possible generalisations to the higher-dimensional torus are explored.

math.AP↗
arXiv2020

Critical Chirality in Elliptic Systems

We establish the regularity in 2 dimensions of $L^2$ solutions to critical elliptic systems in divergence form involving involution operators of finite $W^{1,2}$-energy.

math.AP↗
arXiv2019

3-Commutators Revisited

We present a class of Pseudo-differential elliptic systems with anti-self-dual potentials on ${\mathbb R}$ satisfying compensation phenomena similar to the ones for elliptic systems with anti-symmetric potentials. These compensation phenomena are based on new "multi-commutator" structures generalizing the 3-commtators introduced by the authors in a previous work some years ago.

math.AP↗
arXiv2018

Some Remarks on Pohozaev-Type Identities

The aim of this note is to discuss in more detail the Pohozaev-type identities that have been recently obtained by the author, Paul Laurain and Tristan Rivière in the framework of half-harmonic maps defined either on $R$ or on the sphere $S^1$ with values into a closed manifold $N^n\subset R^m$. Weak half-harmonic maps are critical points of the following nonlocal energy $$\int_{R}|(-Δ)^{1/4}u|^2 dx~~\mbox{or}~~\int_{S^1}|(-Δ)^{1/4}u|^2\ dθ.$$ If $u$ is a sufficiently smooth critical point of the above energy then it satisfies the following equation of stationarity $$\frac{du}{dx}\cdot (-Δ)^{1/2} u=0~~\mbox{a.e in $R$}~~\mbox{or}~~\frac{\partial u}{\partial θ}\cdot (-Δ)^{1/2} u=0~~\mbox{a.e in $S^1$.}$$ By using the invariance of the equation of stationarity in $S^1$ with respect to the trace of the Möbius transformations of the $2$ dimensional disk we derive a countable family of relations involving the Fourier coefficients of weak half-harmonic maps $u\colon S^1\to N^n.$ In the same spirit we also provide as many Pohozaev-type identities in $2$-D for stationary harmonic maps as conformal vector fields in $R^2$ generated by holomorphic functions.

math.AP↗
arXiv2018

A Resolution of the Poisson Problem for Elastic Plates

The Poisson problem consists in finding an immersed surface $Σ\subset\mathbb{R}^m$ minimising Germain's elastic energy (known as Willmore energy in geometry) with prescribed boundary, boundary Gauss map and area which constitutes a non-linear model for the equilibrium state of thin, clamped elastic plates originating from the work of S. Germain and S.D. Poisson or the early XIX century. We present a solution to this problem consisting in the minimisation of the total curvature energy $E(Σ)=\int_Σ|\operatorname{I\!I}_Σ|^2_{g_Σ}\,\mathrm{d}vol_Σ$ ($\operatorname{I\!I}_Σ$ is the second fundamental form of $Σ$), which is variationally equivalent to the elastic energy, in the case of boundary data of class $C^{1,1}$ and when the boundary curve is simple and closed. The minimum is realised by an immersed disk, possibly with a finite number of branch points in its interior, which is of class $C^{1,α}$ up to the boundary for some $0<α<1$, and whose Gauss map extends to a map of class $C^{0,α}$ up to the boundary.

math.DG↗
arXiv2017

Free boundary minimal surfaces: a nonlocal approach

Given a $C^k$-smooth closed embedded manifold $\mathcal N\subset{\mathbb R}^m$, with $k\ge 2$, and a compact connected smooth Riemannian surface $(S,g)$ with $\partial S\neq\emptyset$, we consider $\frac 12$-harmonic maps $u\in H^{1/2}(\partial S,\mathcal N)$. These maps are critical points of the nonlocal energy \begin{equation}E(f;g):=\int_S\big|\nabla\widetilde u\big|^2\,d\text{vol}_g,\end{equation} where $\widetilde u$ is the harmonic extension of $u$ in $S$. We express the energy as a sum of the $\frac 12$-energies at each boundary component of $\partial S$ (suitably identified with the circle $\mathcal S^1$), plus a quadratic term which is continuous in the $H^s(\mathcal S^1)$ topology, for any $s\in\mathbb R$. We show the $C^{k-1,δ}$ regularity of $\frac 12$-harmonic maps. We also establish a connection between free boundary minimal surfaces and critical points of $E$ with respect to variations of the pair $(f,g)$, in terms of the Teichmüller space of $S$.

math.AP↗
arXiv2017

On regularity theory for n/p-harmonic maps into manifolds

In this paper we continue the investigation of the regularity of the so-called weak $\frac{n}{p}$-harmonic maps in the critical case. These are critical points of the following nonlocal energy \[ {\mathcal{L}}_s(u)=\int_{\mathbb{R}^n}| ( {-Δ})^{\frac{s}{2}} u(x)|^p dx\,, \] where $u\in \dot{H}^{s,p}(\mathbb{R}^n,\mathcal{N})$ and ${\mathcal{N}}\subset\mathbb{R}^N$ is a closed $k$ dimensional smooth manifold and $s=\frac{n}{p}$. We prove Hölder continuity for such critical points for $p \leq 2$. For $p > 2$ we obtain the same under an additional Lorentz-space assumption. The regularity theory is in the two cases based on regularity results for nonlocal Schrödinger systems with an antisymmetric potential.

math.AP↗
arXiv2017

Remarks on Neumann boundary problems involving Jacobians

In this short note we explore the validity of Wente-type estimates for Neumann boundary problems involving Jacobians. We show in particular that such estimates do not in general hold under the same hypotheses on the data for Dirichlet boundary problems.

math.AP↗
arXiv2016

A Pohozaev-type formula and Quantization of Horizontal Half-Harmonic Maps

In a recent paper the first and the third authors introduced the notion of horizontal α-harmonic map, with respect to a given C^1 planes distribution P_T on all R^m. The goal of this paper is to investigate compactness and quantization properties of sequences of horizontal 1/2- harmonic maps u_k in 1D. The quantization analysis is obtained through a precise asymptotic development of the energy of u_k in the neck regions and a subtle application of new Pohozaev-type formulae.

math.AP↗
arXiv2016

The nonlocal Liouville-type equation in $\mathbb{R}$ and conformal immersions of the disk with boundary singularities

In this paper we perform a blow-up and quantization analysis of the fractional Liouville equation in dimension $1$. More precisely, given a sequence $u_k :\mathbb{R} \to \mathbb{R}$ of solutions to \begin{equation} (-Δ)^\frac{1}{2} u_k =K_ke^{u_k}\quad \text{in }\mathbb{R}, \end{equation} with $K_k$ bounded in $L^\infty$ and $e^{u_k}$ bounded in $L^1$ uniformly with respect to $k$, we show that up to extracting a subsequence $u_k$ can blow-up at (at most) finitely many points $B=\{a_1,\dots, a_N\}$ and either (i) $u_k\to u_\infty$ in $W^{1,p}_{loc}(\mathbb{R}\setminus B)$ and $K_ke^{u_k} \stackrel{*}{\rightharpoondown} K_\infty e^{u_\infty}+ \sum_{j=1}^N πδ_{a_j}$, or (ii) $u_k\to-\infty$ uniformly locally in $\mathbb{R}\setminus B$ and $K_k e^{u_k}\stackrel{*}{\rightharpoondown} \sum_{j=1}^N α_j δ_{a_j}$ with $α_j\ge π$ for every $j$. This result, resting on the geometric interpretation and analysis provided in a recent collaboration of the authors with T. Rivière and on a classical work of Blank about immersions of the disk into the plane, is a fractional counterpart of the celebrated works of Brézis-Merle and Li-Shafrir on the $2$-dimensional Liouville equation, but providing sharp quantization estimates ($α_j=π$ and $α_j\ge π$) which are not known in dimension $2$ under the weak assumption that $(K_k)$ be bounded in $L^\infty$ and is allowed to change sign.

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