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Luca Martinazzi

Publications and source records attributed to Luca Martinazzi.

At least 19 recordsLinked to original sources
arXiv2026

Non-constancy and multiplicity of half-harmonic maps from intervals into the circle

We study one-dimensional half-harmonic maps from the real line into the circle with prescribed exterior data. We show that, for every positive integer $k$, if two disjoint intervals are sufficiently close, there exist at least $k$ distinct non-constant half-harmonic maps with constant exterior data. More generally, we establish a multiplicity result for boundary data with energy below the critical threshold $2\pi$ by introducing a local relative degree and proving corresponding degree-jump estimates. Working on a single interval and for boundary data arising as traces of finite Blaschke products, we investigate the existence and non-existence of energy minimizers in prescribed degree classes.

math.AP↗
arXiv2024

One-dimensional half-harmonic maps into the circle and their degree

Given a half-harmonic map $u\in \dot H^{\frac{1}{2},2}(\mathbb{R},\mathbb{S}^1)$ minimizing the fractional Dirichlet energy under Dirichlet boundary conditions in $\mathbb{R}\setminus I$, we show the existence of a second half-harmonic map, minimizing the fractional Dirichlet energy in a different homotopy class. This is based on the study of the degree of fractional Sobolev maps and a sharp estimate \`a la Brezis-Coron. We give examples showing that it is in general not possible to minimize in every homotopy class and show a contrast with the 2-dimensional case.

math.AP↗
arXiv2024

Gluing instantons \`a la Brezis-Coron in dimension four and the dipole construction

Given a connection $A$ on a $SU(2)$-bundle $P$ over $\mathbb{R}^4$ with finite Yang-Mills energy $YM(A)$ and nonzero curvature $F_A(0)$ at the origin, and given $\rho>0$ small enough, we construct a new connection $\hat A$ on a bundle $\hat P$ of different Chern class ($|c_2(A)-c_2(\hat A)|=8\pi^2$), in such a way that $\hat A$ is gauge equivalent to $A$ in $\mathbb{R}^4\setminus B_\rho(0)$, gauge equivalent to an instanton in a smaller ball $B_{\tau \rho}(0)$, and $$YM(\hat A)\le YM(A)+8\pi^2-\varepsilon_0\rho^4|F_A(0)|^2,$$ where $\tau\in (0.3,0.4)$ and $\varepsilon_0>0$ are universal constant independent of $A$ and $\rho$. Our gluing method is similar in spirit to the one of Brezis-Coron for harmonic maps. We compare it with classical results by Taubes and discuss applications and open problems.

math.DG↗
arXiv2022

Critical points of the Moser-Trudinger functional on closed surfaces

Given a closed Riemann surface $(Σ,g)$ and any positive smooth weight, we use a minmax scheme together with compactness, quantization results and with sharp energy estimates to prove the existence of positive critical points of the functional $$J_{p,β}(u)=\frac{2-p}{2}\left(\frac{p\|u\|_{H^1}^2}{2β} \right)^{\frac{p}{2-p}}-\ln \int_Σ(e^{u_+^p}-1) f dv_g,$$ for every $p\in (1,2)$ and $β>0$, {or} for $p=1$ and $β\in (0,\infty)\setminus 4π\mathbb{N}$. Letting $p\uparrow 2$ we obtain positive critical points of the Moser-Trudinger functional $$F(u):=\int_Σ(e^{u^2}-1)f dv_g$$ constrained to $\mathcal{E}_β:=\left\{v\text{ s.t. }\|v\|_{H^1}^2=β\right\}$ for any $β>0$.

math.AP↗
arXiv2021

Sign-changing blow-up for the Moser-Trudinger equation

Given a sufficiently symmetric domain $Ω\Subset\mathbb{R}^2$, for any $k\in \mathbb{N}\setminus \{0\}$ and $β>4πk$ we construct blowing-up solutions $(u_\varepsilon)\subset H^1_0(Ω)$ to the Moser-Trudinger equation such that as $\varepsilon\downarrow 0$, we have $\|\nabla u_\varepsilon\|_{L^2}^2\to β$, $u_\varepsilon \rightharpoonup u_0$ in $H^1_0$ where $u_0$ is a sign-changing solution of the Moser-Trudinger equation and $u_\varepsilon$ develops $k$ positive spherical bubbles, all concentrating at $0\in Ω$. These $3$ features (lack of quantization, non-zero weak limit and bubble clustering) stand in sharp contrast to the positive case ($u_\varepsilon>0$) studied by the second author and Druet (J. Eur. Math. Soc. (JEMS) 22 (2020)).

math.AP↗
arXiv2020

Normal conformal metrics on $\mathbb{R}^4$ with $Q$-curvature having power-like growth

Answering a question by M. Struwe (Vietnam J. Math. 2020) related to the blow-up behaviour in the Nirenberg problem, we show that the prescribed $Q$-curvature equation $$Δ^2 u=(1-|x|^p)e^{4u}\text{ in }\mathbb{R}^4,\quad Λ:=\int_{\mathbb{R}^4}(1-|x|^p)e^{4u}dx<\infty$$ has normal solutions (namely solutions which can be written in integral form, and hence satisfy $Δu(x) =O(|x|^{-2})$ as $|x|\to \infty$) if and only if $p\in (0,4)$ and $$\left(1+\frac{p}{4}\right)8π^2\le Λ<16π^2.$$ We also prove existence and non-existence results for the positive curvature case, namely for $Δ^2 u=(1+|x|^p)e^{4u}$ in $\mathbb{R}^4$, and discuss some open questions.

math.AP↗
arXiv2019

Gluing metrics with prescribed $Q$-curvature and different asymptotic behaviour in high dimension

We show a new example of blow-up behaviour for the prescribed $Q$-curvature equation in even dimension $6$ and higher, namely given a sequence $(V_k)\subset C^0(\mathbb{R}^{2n})$ suitably converging we construct {for $n\geq 3$} a sequence $(u_k)$ of radially symmetric solutions to the equation $${(-Δ)^n u_k=V_k e^{2n u_k} \quad \text{in }\mathbb{R}^{2n},}$$ with $u_k$ blowing up at the origin \emph{and} on a sphere. We also prove sharp blow-up estimates. This is in sharp contrast with the $4$-dimensional case studied by F. Robert (J. Diff. Eq. 2006).

math.AP↗
arXiv2018

Concentration phenomena for the fractional $Q$-curvature equation in dimension 3 and fractional Poisson formulas

We study the compactness properties of metrics of prescribed fractional $Q$-curvature of order $3$ in $\R^3$. We will use an approach inspired from conformal geometry, seeing a metric on a subset of $\R^3$ as the restriction of a metric on $\R^4_+$ with vanishing fourth-order $Q$-curvature. We will show that a sequence of such metrics with uniformly bounded fractional $Q$-curvature can blow up on a large set (roughly, the zero set of the trace of a nonpositive biharmonic function $Φ$ in $\R^4_+$), in analogy with a $4$-dimensional result of Adimurthi-Robert-Struwe, and construct examples of such behaviour. In doing so, we produce general Poisson-type representation formulas (also for higher dimension), which are of independent interest.

math.AP↗
arXiv2018

The non-local mean-field equation on an interval

We consider the fractional mean-field equation on the interval $I=(-1,1)$ $$(-Δ)^\frac{1}{2} u=ρ\frac{e^{u}}{\int_{I}e^{u}dx},$$ subject to Dirichlet boundary conditions, and prove that existence holds if and only if $ρ<2π$. This requires the study of blowing-up sequences of solutions. We provide a series of tools in particular which can be used (and extended) to higher-order mean field equations of non-local type.

math.AP↗
arXiv2018

Local and nonlocal singular Liouville equations in Euclidean spaces

We study metrics of constant $Q$-curvature in the Euclidean space with a prescribed singularity at the origin, namely solutions to the equation $$(-Δ)^\frac{n}{2}w=e^{nw}-cδ_{0} \text{ on } \mathbb R^n,$$ under a finite volume condition. We analyze the asymptotic behaviour at infinity and the existence of solutions for every $n\ge 3$ also in a supercritical regime. Finally, we state some open problems.

math.AP↗
arXiv2017

The Moser-Trudinger inequality and its extremals on a disk via energy estimates

We study the Dirichlet energy of non-negative radially symmetric critical points $u_μ$ of the Moser-Trudinger inequality on the unit disc in $\mathbb{R}^2$, and prove that it expands as $$4π+\frac{4π}{μ^{4}}+o(μ^{-4})\le \int_{B_1}|\nabla u_μ|^2dx\le 4π+\frac{6π}{μ^{4}}+o(μ^{-4}),\quad \text{as }μ\to\infty,$$ where $μ=u_μ(0)$ is the maximum of $u_μ$. As a consequence, we obtain a new proof of the Moser-Trudinger inequality, of the Carleson-Chang result about the existence of extremals, and of the Struwe and Lamm-Robert-Struwe multiplicity result in the supercritical regime (only in the case of the unit disk). Our results are stable under sufficiently weak perturbations of the Moser-Trudinger functional. We explicitly identify the critical level of perturbation for which, although the perturbed Moser-Trudinger inequality still holds, the energy of its critical points converges to $4π$ from below. We expect, in some of these cases, that the existence of extremals does not hold, nor the existence of critical points in the supercritical regime.

math.AP↗
arXiv2016

Large blow-up sets for the prescribed Q-curvature equation in the Euclidean space

Let $m\ge 2$ be an integer. For any open domain $Ω\subset\mathbb{R}^{2m}$, non-positive function $φ\in C^\infty(Ω)$ such that $Δ^m φ\equiv 0$, and bounded sequence $(V_k)\subset L^\infty(Ω)$ we prove the existence of a sequence of functions $(u_k)\subset C^{2m-1}(Ω)$ solving the Liouville equation of order $2m$ $$(-Δ)^m u_k = V_ke^{2mu_k}\quad \text{in }Ω, \quad \limsup_{k\to\infty} \int_Ωe^{2mu_k}dx<\infty,$$ and blowing up exactly on the set $S_φ:=\{x\in Ω:φ(x)=0\}$, i.e. $$\lim_{k\to\infty} u_k(x)=+\infty \text{ for }x\in S_φ \text{ and }\lim_{k\to\infty} u_k(x)=-\infty \text{ for }x\in Ω\setminus S_φ,$$ thus showing that a result of Adimurthi, Robert and Struwe is sharp. We extend this result to the boundary of $Ω$ and to the case $Ω=\mathbb{R}^{2m}$. Several related problems remain open.

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.

math.DG↗
arXiv2015

Fractional Adams-Moser-Trudinger type inequalities

Extending several works, we prove a general Adams-Moser-Trudinger type inequality for the embedding of Bessel-potential spaces $\tilde H^{\frac{n}{p},p}(Ω)$ into Orlicz spaces for an arbitrary domain $Ω\subset \mathbb{R}^n$ with finite measure. In particular we prove $$\sup_{u\in \tilde H^{\frac{n}{p},p}(Ω), \;\|(-Δ)^{\frac{n}{2p}}u\|_{L^{p}(Ω)}\leq 1}\int_Ωe^{α_{n,p} |u|^\frac{p}{p-1}}dx \leq c_{n,p}|Ω|, $$ for a positive constant $α_{n,p}$ whose sharpness we also prove. We further extend this result to the case of Lorentz-spaces (i.e. $(-Δ)^\frac{n}{2p}u\in L^{(p,q)})$. The proofs are simple, as they use Green functions for fractional Laplace operators and suitable cut-off procedures to reduce the fractional results to the sharp estimate on the Riesz potential proven by Adams and its generalization proven by Xiao and Zhai. We also discuss an application to the problem of prescribing the $Q$-curvature and some open problems.

math.AP↗
arXiv2015

Blow-up behaviour of a fractional Adams-Moser-Trudinger type inequality in odd dimension

Given a smoothly bounded domain $Ω\Subset\mathbb{R}^n$ with $n\ge 1$ odd, we study the blow-up of bounded sequences $(u_k)\subset H^\frac{n}{2}_{00}(Ω)$ of solutions to the non-local equation $$(-Δ)^\frac n2 u_k=λ_k u_ke^{\frac n2 u_k^2}\quad \text{in }Ω,$$ where $λ_k\toλ_\infty \in [0,\infty)$, and $H^{\frac n2}_{00}(Ω)$ denotes the Lions-Magenes spaces of functions $u\in L^2(\mathbb{R}^n)$ which are supported in $Ω$ and with $(-Δ)^\frac{n}{4}u\in L^2(\mathbb{R}^n)$. Extending previous works of Druet, Robert-Struwe and the second author, we show that if the sequence $(u_k)$ is not bounded in $L^\infty(Ω)$, a suitably rescaled subsequence $η_k$ converges to the function $η_0(x)=\log\left(\frac{2}{1+|x|^2}\right)$, which solves the prescribed non-local $Q$-curvature equation $$(-Δ)^\frac n2 η=(n-1)!e^{nη}\quad \text{in }\mathbb{R}^n$$ recently studied by Da Lio-Martinazzi-Rivière when $n=1$, Jin-Maalaoui-Martinazzi-Xiong when $n=3$, and Hyder when $n\ge 5$ is odd. We infer that blow-up can occur only if $Λ:=\limsup_{k\to \infty}\|(-Δ)^\frac n4 u_k\|_{L^2}^2\ge Λ_1:= (n-1)!|S^n|$.

math.AP↗