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Maria del Mar Gonzalez

Publications and source records attributed to Maria del Mar Gonzalez.

At least 19 recordsLinked to original sources
arXiv2025

The limiting case of the fractional Caffarelli-Kohn-Nirenberg inequality in dimension one

In this paper we study the fractional Caffarelli-Kohn-Nirenberg inequality (CKN) in one dimension when the parameter $γ$ converges (from the left) to its critical value $1/2$, obtaining Onofri's inequality in the unit disk as the limit. A difficulty that we encounter is the lack of an explicit expression for the extremal function at which the CKN inequality is attained, which we address by studying solutions of the weighted Liouville equation for the half-Laplacian in dimension one.

math.AP↗
arXiv2022

Deterministic Decoupling of Global Features and its Application to Data Analysis

We introduce a method for deterministic decoupling of global features and show its applicability to improve data analysis performance, as well as to open new venues for feature transfer. We propose a new formalism that is based on defining transformations on submanifolds, by following trajectories along the features gradients. Through these transformations we define a normalization that, we demonstrate, allows for decoupling differentiable features. By applying this to sampling moments, we obtain a quasi-analytic solution for the orthokurtosis, a normalized version of the kurtosis that is not just decoupled from mean and variance, but also from skewness. We apply this method in the original data domain and at the output of a filter bank to regression and classification problems based on global descriptors, obtaining a consistent and significant improvement in performance as compared to using classical (non-decoupled) descriptors.

cs.LG↗
arXiv2022

Symmetry and symmetry breaking for the fractional Caffarelli-Kohn-Nirenberg inequality

In this paper, we will consider the fractional Caffarelli-Kohn-Nirenberg inequality \begin{equation*} Λ \left(\int_{\mathbb R^n}\frac{|u(x)|^{p}}{|x|^{β {p}}}\,dx\right)^{\frac{2}{p}}\leq \int_{\mathbb R^n}\int_{\mathbb R^n}\frac{(u(x)-u(y))^2}{|x-y|^{n+2γ}|x|^{α}|y|^{α}}\,dy\,dx \end{equation*} where $γ\in(0,1)$, $n\geq 2$, and $α,β\in\mathbb R$ satisfy \begin{equation*} α\leq β\leq α+γ, \ -2γ<α<\frac{n-2γ}{2}, \end{equation*} and the exponent $p$ is chosen to be \begin{equation*} p=\frac{2n}{n-2γ+2(β-α)}, \end{equation*} such that the inequality is invariant under scaling. We first study the existence and nonexistence of extremal solutions. Our next goal is to show some results on the symmetry and symmetry breaking region for the minimizers; these suggest the existence of a Felli-Schneider type curve separating both regions but, surprisingly, we find a novel behavior as $α\to -2γ$. The main idea in the proofs, as in the classical case, is to reformulate the fractional Caffarelli-Kohn-Nirenberg inequality in cylindrical variables. Then, in order to find the radially symmetric solutions we need to solve a non-local ODE. For this equation we also get uniqueness of minimizers in the radial symmetry class; indeed, we show that the unique continuation argument of Frank-Lenzmann (Acta'13) can be applied to more general operators with good spectral properties. We provide, in addition, a completely new proof of non-degeneracy which works for all critical points. It is based on the variation of constants approach and the non-local Wronskian of Ao-Chan-DelaTorre-Fontelos-González-Wei (Duke'19).

math.AP↗
arXiv2021

Eigenvalue bounds for the Paneitz operator and its associated third-order boundary operator on locally conformally flat manifolds

In this paper we study bounds for the first eigenvalue of the Paneitz operator $P$ and its associated third-order boundary operator $B^3$ on four-manifolds. We restrict to orientable, simply connected, locally confomally flat manifolds that have at most two umbilic boundary components. The proof is based on showing that under the hypotheses of the main theorems, the considered manifolds are confomally equivalent to canonical models. This equivalence is proved by showing the injectivity of suitable developing maps. Then the bounds on the eigenvalues are obtained through explicit computations on the canonical models and its connections with the classes of manifolds that we are considering. The fact that $P$ and $B^3$ are conformal in four dimensions is key in the proof.

math.DG↗
arXiv2020

Removability of singularities and superharmonicity for some fractional Laplacian equations

We study some qualitative properties (including removable singularities and superharmonicity) of non-negative solutions to $$ (-Δ)^γu=fu^p\quad\text{in }\mathbb R^n\setminusΣ$$ which are singular at $Σ$. Here $γ\in (0, \frac{n}{2})$. Among other things, we first prove that if $Σ$ is a compact set in $\mathbb R^n$ with Assouad dimension $\bf d$ (not necessarily an integer), ${\bf d} \frac{n-\bf d}{n-{\bf d}-2γ},$$ then $u\in L^p_{loc}(\mathbb R^n)$ and $u$ is a distributional solution in $\mathbb R^n$. Then we prove that $ (-Δ)^σu >0$ for all $ σ\in (0, γ)$, if $Σ=ϕ$.

math.AP↗
arXiv2019

On higher dimensional singularities for the fractional Yamabe problem: a non-local Mazzeo-Pacard program

We consider the problem of constructing solutions to the fractional Yamabe problem that are singular at a given smooth sub-manifold, and we establish the classical gluing method of Mazzeo and Pacard for the scalar curvature in the fractional setting. This proof is based on the analysis of the model linearized operator, which amounts to the study of an ODE, and thus our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of non-local ODEs. No traditional phase-plane analysis is available here. Instead, first, we provide a rigorous construction of radial fast-decaying solutions by a blow-up argument and a bifurcation method. Second, we use conformal geometry to rewrite this non-local ODE, giving a hint of what a non-local phase-plane analysis should be. Third, for the linear theory, we examine a fractional Schrödinger equation with a Hardy type critical potential. We construct its Green's function, deduce Fredholm properties, and analyze its asymptotics at the singular points in the spirit of Frobenius method. Surprisingly enough, a fractional linear ODE may still have a two-dimensional kernel as in the second order case.

math.AP↗
arXiv2019

Bound state solutions for the supercritical fractional Schrödinger equation

We prove the existence of positive solutions for the supercritical nonlinear fractional Schrödinger equation $(-Δ)^s u+V(x)u-u^p=0$ in $\mathbb R^n$, with $u(x)\to 0$ as $|x|\to +\infty$, where $p>\frac{n+2s}{n-2s}$ for $s\in (0,1), \ n>2s$. We show that if $V(x)=o(|x|^{-2s})$ as $|x|\to +\infty$, then for $p>\frac{n+2s-1}{n-2s-1}$, this problem admits a continuum of solutions. More generally, for $p>\frac{n+2s}{n-2s}$, conditions for solvability are also provided. This result is the extension of the work by Davila, Del Pino, Musso and Wei to the fractional case. Our main contributions are: the existence of a smooth, radially symmetric, entire solution of $(-Δ)^s w=w^p$ in $\mathbb R^n$, and the analysis of its properties. The difficulty here is the lack of phase-plane analysis for a nonlocal ODE; instead we use conformal geometry methods together with Schaaf's argument as in the paper by Ao, Chan, DelaTorre, Fontelos, González and Wei on the singular fractional Yamabe problem.

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

A gluing approach for the fractional Yamabe problem with isolated singularities

We construct solutions for the fractional Yamabe problem that are singular at a prescribed number of isolated points. This seems to be the first time that a gluing method is successfully applied to a non-local problem. The main step is an infinite-dimensional Lyapunov-Schmidt reduction method, that reduces the problem to an (infinite dimensional) Toda type system.

math.AP↗
arXiv2018

Boundary connected sum of Escobar manifolds

Let $(X_1, \bar g_1)$ and $(X_2, \bar g_2)$ be two compact Riemannian manifolds with boundary $(M_1,g_1)$ and $(M_2,g_2)$ respectively. The Escobar problem consists in prescribing a conformal metric on a compact manifold with boundary with zero scalar curvature in the interior and constant mean curvature of the boundary. The present work is the construction of a connected sum $X=X_1 \sharp X_2$ by excising half ball near points on the boundary. The resulting metric on $X$ has zero scalar curvature and a CMC boundary. We fully exploit the nonlocal aspect of the problem and use new tools developed in recent years to handle such kinds of issues. Our problem is of course a very well-known problem in geometric analysis and that is why we consider it but the results in the present paper can be extended to other more analytical problems involving connected sums of constant fractional curvatures.

math.DG↗
arXiv2016

Fractional Laplacians and extension problems: the higher rank case

The aim of this paper is two-fold: first, we look at the fractional Laplacian and the conformal fractional Laplacian from the general framework of representation theory on symmetric spaces and, second, we construct new boundary operators with good conformal properties that generalize the fractional Laplacian using an extension problem in which the boundary is of codimension two.

math.AP↗
arXiv2015

A nonlocal diffusion problem on manifolds

In this paper we study a nonlocal diffusion problem on a manifold. These kind of equations can model diffusions when there are long range effects and have been widely studied in Euclidean space. We first prove existence and uniqueness of solutions and a comparison principle. Then, for a convenient rescaling we prove that the operator under consideration converges to a multiple of the usual Heat-Beltrami operator on the manifold. Next, we look at the long time behavior on compact manifolds by studying the spectral properties of the operator. Finally, for the model case of hyperbolic space we study the long time asymptotics and find a different and interesting behavior.

math.AP↗
arXiv2015

Asymptotic behavior of Palais-Smale sequences associated with fractional Yamabe type equations

In this paper, we analyze the asymptotic behavior of Palais-Smale sequences associated with fractional Yamabe type equations on an asymptotically hyperbolic Riemannian manifold. We prove that Palais-Smale sequences can be decomposed into the solution of the limit equation plus a finite number of bubbles, which are the rescaling of the fundamental solutions to the fractional Yamabe equation on Euclidean space. We also verify the non-interfering fact for multi-bubbles.

math.AP↗
arXiv2015

Further results on the fractional Yamabe problem: the umbilic case

We prove some existence results for the fractional Yamabe problem in the case that the boundary manifold is umbilic, thus covering some of the cases not considered by Gonzalez and Qing. These are inspired by the work of Coda-Marques on the boundary Yamabe problem but, in addition, a careful understanding of the behavior at infinity for asymptotically hyperbolic metrics is required.

math.AP↗
arXiv2013

Fractional conformal Laplacians and fractional Yamabe problems

Based on the relations between scattering operators of asymptotically hyperbolic metrics and Dirichlet-to-Neumann operators of uniformly degenerate elliptic boundary value problems, we formulate fractional Yamabe problems that include the boundary Yamabe problem studied by Escobar. We observe an interesting Hopf type maximum principle together with interplays between analysis of weighted trace Sobolev inequalities and conformal structure of the underlying manifolds, which extend the phenomena displayed in the classic Yamabe problem and boundary Yamabe problem.

math.DG↗
arXiv2012

A discrete Bernoulli free boundary problem

We consider a free boundary problem for the $p$-Laplace operator which is related to the so-called Bernoulli free boundary problem. In this formulation, the classical boundary gradient condition is replaced by a condition on the distance between two different level surfaces of the solution. For suitable scalings our model converges to the classical Bernoulli problem; one of the advantages in this new formulation lies in the simplicity of the arguments, since one does not need to consider the boundary gradient. We shall study this problem in convex and other regimes, and establish existence and qualitative theory.

math.AP↗