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Luca Martinazzi

Publications and source records attributed to Luca Martinazzi.

36 records · Page 2Linked to original sources
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↗
arXiv2015

Blow-up analysis of a nonlocal Liouville-type equation

In this paper we perform a blow-up and quantization analysis of the following nonlocal Liouville-type equation \begin{equation}(-Δ)^\frac12 u= κe^u-1~\mbox{in $S^1$,} \end{equation} where $(-Δ)^\frac{1}{2}$ stands for the fractional Laplacian and $κ$ is a bounded function. We interpret the above equation as the prescribed curvature equation to a curve in conformal parametrization. We also establish a relation between this equation and the analogous equation in $\mathbb{R}$ \begin{equation} (-Δ)^\frac{1}{2} u =Ke^u \quad \text{in }\mathbb{R}, \end{equation} with $K$ bounded on $\mathbb{R}$.

math.DG↗
arXiv2014

Existence and asymptotics for solutions of a non-local Q-curvature equation in dimension three

We study conformal metrics on $R^3$, i.e., metrics of the form $g_u=e^{2u}|dx|^2$, which have constant $Q$-curvature and finite volume. This is equivalent to studying the non-local equation $$ (-Δ)^\frac32 u = 2 e^{3u}$$ in $R^3$ $$V:=\int_{\mathbb{R}^3}e^{3u}dx<\infty,$$ where $V$ is the volume of $g_u$. Adapting a technique of A. Chang and W-X. Chen to the non-local framework, we show the existence of a large class of such metrics, particularly for $V\le 2π^2=|S^3|$. Inspired by previous works of C-S. Lin and L. Martinazzi, who treated the analogue cases in even dimensions, we classify such metrics based on their behavior at infinity.

math.DG↗
arXiv2014

Conformal metrics on $R^{2m}$ with constant Q-curvature, prescribed volume and asymptotic behavior

We study the solutions $u\in C^\infty(R^{2m})$ of the problem $(-Δ)^m u= Qe^{2mu}$, where $Q=\pm (2m-1)!$, and $V :=\int_{R^{2m}}e^{2mu}dx <\infty$, particularly when $m>1$. This corresponds to finding conformal metrics $g_u:=e^{2u}|dx|^2$ on $R^{2m}$ with constant Q-curvature $Q$ and finite volume $V$. Extending previous works of Chang-Chen, and Wei-Ye, we show that both the value $V$ and the asymptotic behavior of $u(x)$ as $|x|\to \infty$ can be simultaneously prescribed, under certain restrictions. When $Q=(2m-1)!$ we need to assume $V<vol(S^{2m})$, but surprisingly for $Q=-(2m-1)!$ the volume $V$ can be chosen arbitrarily.

math.DG↗
arXiv2012

Critical points of the Moser-Trudinger functional on a disk

On the 2-dimensional unit disk $B_1$ we study the Moser-Trudinger functional $$E(u)=\int_{B_1}(e^{u^2}-1)dx, u\in H^1_0(B_1)$$ and its restrictions to $M_Λ:=\{u \in H^1_0(B_1):\|u\|^2_{H^1_0}=Λ\}$ for $Λ>0$. We prove that if a sequence $u_k$ of positive critical points of $E|_{M_{Λ_k}}$ (for some $Λ_k>0$) blows up as $k\to\infty$, then $Λ_k\to 4π$, and $u_k\to 0$ weakly in $H^1_0(B_1)$ and strongly in $C^1_{\loc}(\bar B_1\setminus\{0\})$. Using this we also prove that when $Λ$ is large enough, then $E|_{M_Λ}$ has no positive critical point, complementing previous existence results by Carleson-Chang, M. Struwe and Lamm-Robert-Struwe.

math.FA↗
arXiv2012

Conformal metrics on R^{2m} with constant Q-curvature and large volume

We study conformal metrics on R^{2m} with constant Q-curvature and finite volume. When m=3 we show that there exists V* such that for any V\in [V*,\infty) there is a conformal metric g on R^{6} with Q_g = Q-curvature of S^6, and vol(g)=V. This is in sharp contrast with the four-dimensional case, treated by C-S. Lin. We also prove that when $m$ is odd and greater than 1, there is a constant V_m>\vol (S^{2m}) such that for every V\in (0,V_m] there is a conformal metric g on R^{2m} with Q_g = Q-curvature of S^{2m}, vol(g)=V. This extends a result of A. Chang and W-X. Chen. When m is even we prove a similar result for conformal metrics of negative Q-curvature.

math.DG↗
arXiv2011

Critical points of the Moser-Trudinger functional

On a smooth bounded 2-dimensional domain $Ω$ we study the heat flow $u_t=Δu +λ(t)ue^{u^2}$ ($λ(t)$ is such that $d/dt ||u(t,\cdot)||_{H^1_0}=0$) introduced by T. Lamm, F. Robert and M. Struwe to investigate the Moser-Trudinger functional $E(v)=\int_Ω (e^{v^2}-1)dx, v\in H^1_0(Ω).$ We prove that if $u$ blows-up as $t\to\infty$ and if $E(u(t,\cdot))$ remains bounded, then for a sequence $t_k\to\infty$ we have $u(t_k,\cdot)\rightharpoonup 0$ in $H^1_0$ and $\|u(t_k,\cdot)\|_{H^1_0}^2\to 4πL$ for an integer $L\ge 1$. We couple these results with a topological technique to prove that if $Ω$ is not contractible, then for every $0<Λ\in \mathbb{R} \setminus 4 π\mathbb{N}$ the functional $E$ constrained to $M_Λ=\{v\in H^1_0(Ω):||v||_{H^1_0}^2=Λ\}$ has a positive critical point. We prove that when $Ω$ is the unit ball and $Λ$ is large enough, then $E|_{M_Λ}$ has no positive critical points, hence showing that the topological assumption on $Ω$ is natural.

math.FA↗
arXiv2010

Quantization for the prescribed Q-curvature equation on open domains

We discuss compactness, blow-up and quantization phenomena for the prescribed $Q$-curvature equation $(-Δ)^m u_k=V_ke^{2mu_k}$ on open domains of $\R{2m}$. Under natural integral assumptions we show that when blow-up occurs, up to a subsequence $$\lim_{k\to \infty}\int_{Ω_0} V_ke^{2mu_k}dx=LΛ_1,$$ where $Ω_0\subset\subsetΩ$ is open and contains the blow-up points, $L\in\mathbb{N}$ and $Λ_1:=(2m-1)!\vol(S^{2m})$ is the total $Q$-curvature of the round sphere $S^{2m}$. Moreover, under suitable assumptions, the blow-up points are isolated. We do not assume that $V$ is positive.

math.AP↗
arXiv2010

Quantization for an elliptic equation of order 2m with critical exponential non-linearity

On a smoothly bounded domain $Ω\subset\R{2m}$ we consider a sequence of positive solutions $u_k\stackrel{w}{\rightharpoondown} 0$ in $H^m(Ω)$ to the equation $(-Δ)^m u_k=λ_k u_k e^{mu_k^2}$ subject to Dirichlet boundary conditions, where $0<λ_k\to 0$. Assuming that $$Λ:=\lim_{k\to\infty}\int_Ωu_k(-Δ)^m u_k dx<\infty,$$ we prove that $Λ$ is an integer multiple of $Λ_1:=(2m-1)!\vol(S^{2m})$, the total $Q$-curvature of the standard $2m$-dimensional sphere.

math.FA↗
arXiv2009

Asymptotics and quantization for a mean-field equation of higher order

Given a regular bounded domain $Ω\subset\R{2m}$, we describe the limiting behavior of sequences of solutions to the mean field equation of order $2m$, $m\geq 1$, $$(-Δ)^m u=ρ\frac{e^{2mu}}{\int_Ωe^{2mu}dx}\quad\text{in}Ω,$$ under the Dirichlet boundary condition and the bound $0<ρ\leq C$. We emphasize the connection with the problem of prescribing the $Q$-curvature.

math.AP↗
arXiv2009

A threshold phenomenon for embeddings of $H^m_0$ into Orlicz spaces

We consider a sequence of positive smooth critical points of the Adams-Moser-Trudinger embedding of $H^m_0$ into Orlicz spaces. We study its concentration-compactness behavior and show that if the sequence is not precompact, then the liminf of the $H^m_0$-norms of the functions is greater than or equal to a positive geometric constant.

math.AP↗
arXiv2009

Concentration-compactness phenomena in the higher order Liouville's equation

We investigate different concentration-compactness phenomena related to the Q-curvature in arbitrary even dimension. We first treat the case of an open domain in $R^{2m}$, then that of a closed manifold and, finally, the particular case of the sphere $S^{2m}$. In all cases we allow the sign of the Q-curvature to vary, and show that in the case of a closed manifold, contrary to the case of open domains in $R^{2m}$, concentration phenomena can occur only at points of positive Q-curvature. As a consequence, on a locally conformally flat manifold of non-positive Euler characteristic we always have compactness.

math.AP↗
arXiv2008

Conformal metrics on $\R^{2m}$ with constant Q-curvature

We study the conformal metrics on $\R^{2m}$ with constant Q-curvature $Q$ having finite volume, particularly in the case $Q\leq 0$. We show that when $Q<0$ such metrics exist in $\R^{2m}$ if and only if $m>1$. Moreover we study their asymptotic behavior at infinity, in analogy with the case $Q>0$, which we treated in a recent paper. When Q=0, we show that such metrics have the form $e^{2p}g_{\R^{2m}}$, where $p$ is a polynomial such that $2\leq °p\leq 2m-2$ and $\sup_{\R^{2m}}p<+\infty$. In dimension 4, such metrics are exactly the polynomials $p$ of degree 2 with $\lim_{|x|\to+\infty}p(x)=-\infty$.

math.AP↗
arXiv2008

Classification of solutions to the higher order Liouville's equation on R^{2m}

We classify the solutions to the equation (- Δ)^m u=(2m-1)!e^{2mu} on R^{2m} giving rise to a metric g=e^{2u}g_{R^{2m}} with finite total $Q$-curvature in terms of analytic and geometric properties. The analytic conditions involve the growth rate of u and the asymptotic behaviour of Δu(x) as |x|\to \infty. As a consequence we give a geometric characterization in terms of the scalar curvature of the metric e^{2u}g_{R^{2m}} at infinity, and we observe that the pull-back of this metric to $S^{2m}$ via the stereographic projection can be extended to a smooth Riemannian metric if and only if it is round.

math.AP↗