arXiv subjects

Juan Carlos Fernández

Publications and source records attributed to Juan Carlos Fernández.

12 recordsLinked to original sources

arXiv2026

The conformal logarithmic Laplacian on the sphere: Yamabe-type problems and Sobolev spaces

We study the conformal logarithmic Laplacian on the sphere, an explicit singular integral operator that arises as the derivative (with respect to the order parameter) of the conformal fractional Laplacian at zero. Our analysis provides a detailed investigation of its spectral properties, its conformal invariance, and the associated $Q$-curvature problem. Furthermore, we establish a precise connection between this operator on the sphere and the logarithmic Laplacian in $\mathbb{R}^N$ via stereographic projection. This correspondence bridges classification results for two Yamabe-type problems previously studied in the literature, extending one of them to the weak setting. To this end, we introduce a Hilbert space that serves as the logarithmic counterpart of the homogeneous fractional Sobolev space, offering a natural functional framework for the variational study of logarithmic-type equations in unbounded domains.

math.AP↗

arXiv2026

Infinitely many solutions to Kirchhoff-Boussinesq-type equations on Riemannian manifolds

We study Kirchhoff-Boussinesq-type equations on closed Riemannian manifolds $(M,g)$ of the form \[ \sum_{j=0}^m a_i(-Δ_g)^m u \pm \text{div}_g(\vert\nabla u\vert_g^{p-2}\nabla u) = f(u),\qquad \text{ on }\ M, \] where $m<\dim M/2$, $(a_0,a_1,\ldots a_m)\in C^\infty_+(M)\times [0,\infty)^{m-1}\times(0,\infty)$, $2<p\leq\frac{2\dim M}{\dim M- 2}$ and $f:\mathbb{R}\rightarrow\mathbb{R}$ is a continuous nonlinearity of superlinear type with subcritical or critical Sobolev growth. We briefly motivate the above equation in the case of $m=2$, as an extension of the stationary Kirchhoff-Boussinesq equation when modeling the dynamics of curved elastic plates. Under some symmetry assumptions, we prove the existence of multiple equivariant solutions when considering critical Sobolev nonlinearities and, for subcritical ones, we also prove the existence of ground-state solutions. As a byproduct, we prove a Gagliardo-Nirenberg interpolation inequality and give several equivalent norms on the higher order Sobolev space $H_g^m(M)$.

math.AP↗

arXiv2026

A novel ordinal multi-view aggregation scheme for oak defoliation

Forest decline driven by climate and biotic stressors threatens ecosystem functioning, making accurate monitoring of tree health essential. In this work, we address tree defoliation estimation as an ordinal classification problem using ground-level imagery. We propose a novel multi-view ensemble framework that aggregates predictions from Convolutional Neural Networks (CNNs) trained on different perspectives of individual trees (north, south, and crown). This approach leverages complementary visual information while preserving modelling consistency through a homogeneous ensemble design. A comprehensive evaluation is conducted by comparing multiple ordinal classification methods and analysing the contribution of each view and their combinations. Results show that modelling the ordinal structure of defoliation levels improves performance over nominal approaches, while the proposed multi-view ensemble consistently outperforms single-view and pairwise configurations. In particular, the three-view ensemble achieves the most robust and accurate predictions across all evaluation metrics. These findings highlight the potential of combining Deep Learning (DL), Ordinal Classification (OC), and multi-view aggregation for scalable, consistent, and objective forest health assessment in complex ecosystems such as Mediterranean dehesas.

cs.CV↗

arXiv2025

Fractional $Q$-curvature on the sphere and optimal partitions

We study an optimal partition problem on the sphere, where the cost functional is associated with the fractional $Q$-curvature in terms of the conformal fractional Laplacian on the sphere. By leveraging symmetries, we prove the existence of a symmetric minimal partition through a variational approach. A key ingredient in our analysis is a new Hölder regularity result for symmetric functions in a fractional Sobolev space on the sphere. As a byproduct, we establish the existence of infinitely many solutions to a nonlocal weakly-coupled competitive system on the sphere that remain invariant under a group of conformal diffeomorphisms and we investigate the asymptotic behavior of least-energy solutions as the coupling parameters approach negative infinity.

math.AP↗

arXiv2024

A geometric reduction method for some fully nonlinear first-order PDEs on semi-Riemannian manifolds

Given a semi-Riemannian manifold $(M,\langle \cdot,\cdot\rangle_g),$ we use the transnormal functions defined on $M$ to reduce fully nonlinear first order PDEs of the form \[ F(x,u,\langle \nabla_g u, \nabla_g u \rangle_g) = 0,\qquad \text{on }M \] into ODEs and obtain local existence results of solutions which are constant along the level sets of the transnormal functions. In particular, we apply this reduction method to obtain new solutions to eikonal equations with a prescribed geometry.

math.AP↗

arXiv2023

Equivariant solutions to the optimal partition problem for the prescribed Q-curvature equation

We study the optimal partition problem for the prescribed constant $Q$-curvature equation induced by the higher order conformal operators under the effect of cohomogeneity one actions on Einstein manifolds with positive scalar curvature. This allows us to give a precise description of the solution domains and their boundaries in terms of the orbits of the action. We also prove the existence of least energy symmetric solutions to a weakly coupled elliptic system of prescribed $Q$-curvature equations under weaker assumptions and conclude a multiplicity result of sign-changing solutions to the prescribed constant $Q$-curvature problem induced by the Paneitz-Branson operator. Moreover, we study the coercivity of $GJMS$-operators on Ricci solitons, compute the $Q$-curvature of these manifolds, and give a multiplicity result for the sign-changing solutions to the Yamabe problem with prescribed number of nodal domains on the Koiso-Cao Ricci soliton.

math.AP↗

arXiv2022

Yamabe problem in the presence of singular Riemannian Foliations

Using variational methods together with symmetries given by singular Riemannian foliations with positive dimensional leaves, we prove the existence of an infinite number of sign-changing solutions to Yamabe type problems, which are constant along the leaves of the foliation, and one positive solution of minimal energy among any other solution with these symmetries. In particular, we find sign-changing solutions to the Yamabe problem on the round sphere with new qualitative behavior when compared to previous results, that is, these solutions are constant along the leaves of a singular Riemannian foliation which is not induced neither by a group action nor by an isoparametric function. To prove the existence of these solutions, we prove a Sobolev embedding theorem for general singular Riemannian foliations, and a Principle of Symmetric Criticality for the associated energy functional to a Yamabe type problem.

math.AP↗

arXiv2021

Critical polyharmonic systems and optimal partitions

We establish the existence of solutions to a weakly-coupled competitive system of polyharmonic equations in R^N which are invariant under a group of conformal diffeomorphisms, and study the behavior of least energy solutions as the coupling parameters tend to $-\infty$. We show that the supports of the limiting profiles of their components are pairwise disjoint smooth domains and solve a nonlinear optimal partition problem of R^N. We give a detailed description of the shape of these domains.

math.AP↗

arXiv2020

An ODE reduction method for the semi-Riemannian Yamabe problem on space forms

We consider the semi-Riemannian Yamabe type equations of the form \[ -\square u + λu = μ\vert u\vert^{p-1}u\quad\text{ on }M \] where $M$ is either the semi-Euclidean space or the pseudosphere of dimension $m\geq 3$, $\square$ is the semi-Riemannian Laplacian in $M$, $λ\geq0$, $μ\in\mathbb{R}\smallsetminus\{0\}$ and $p>1$. Using semi-Riemannian isoparametric functions on $M$, we reduce the PDE into a generalized Emden-Fowler ODE of the form \[ w''+q(r)w'+λw = μ\vert w\vert^{p-1}w\quad\text{ on } I, \] where $I\subset\mathbb{R}$ is $[0,\infty)$ or $[0,π]$, $q(r)$ blows-up at $0$ and $w$ is subject to the natural initial conditions $w'(0)=0$ in the first case and $w'(0)=w'(π)=0$ in the second. We prove the existence of blowing-up and globally defined solutions to this problem, both positive and sign-changing, inducing solutions to the semi-Riemannian Yamabe type problem with the same qualitative properties, with level and critical sets described in terms of semi-Riemannian isoparametric hypersurfaces and focal varieties. In particular, we prove the existence of sign-changing blowing-up solutions to the semi-Riemannian Yamabe problem in the pseudosphere having a prescribed number of nodal domains.

math.AP↗

arXiv2019

Low energy nodal solutions to the Yamabe equation

Given an isoparametric function $f$ on the $n$-dimensional sphere, we consider the space of functions $w\circ f$ to reduce the Yamabe equation on the round sphere into a singular ODE on $w$ in the interval $[0,π]$, of the form $w" + (h(r)/\sin r)w'+λ(\vert w\vert^{4/n-2}w - w)=0$, where $h$ is a monotone function with exactly one zero on $[0,π]$ and $λ>0$ is a constant. The natural boundary conditions in order to obtain smooth solutions are $w'(0)=0$ and $w'(π)=0$. We show that for any positive integer $k$ there exists a solution with exactly $k$-zeroes yielding solutions to the Yamabe equation with exactly $k$ connected isoparametric hypersurfaces as nodal set. The idea of the proof is to consider the initial value problems on both singularities $0$ and $π$, and then to solve the corresponding double shooting problem, matching the values of $w$ and $w'$ at the unique zero of $h$. In particular we obtain solutions with exactly one zero, providing solutions of the Yamabe equation with low energy, which can be computed easily by numerical methods.

math.AP↗

arXiv2019

Supercritical elliptic problems on the round sphere and nodal solutions to the Yamabe problem in projective spaces

Given an isoparametric function $f$ on the $n$-dimensional round sphere, we consider functions of the form $u=w\circ f$ to reduce the semilinear elliptic problem \[ -Δ_{g_0}u+λu=λ | u\ | ^{p-1}u\qquad\text{ on }\mathbb{S}^n \] with $λ>0$ and $1 \frac{n+2}{n-2}$, i.e., in the supercritical case. Moreover, using a reduction via harmonic morphisms, we prove existence and multiplicity of sign-changing solutions to the Yamabe problem on the complex and quaternionic space, having a finite disjoint union of isoparametric hipersurfaces as regular level sets.

math.AP↗

arXiv2017

Multiplicity of nodal solutions to the Yamabe problem

Given a compact Riemannian manifold $(M,g)$ without boundary of dimension $m\geq 3$ and under some symmetry assumptions, we establish existence of one positive and multiple nodal solutions to the Yamabe-type equation $$-div_{g}(a\nabla u)+bu=c|u|^{2^{\ast}-2}u\quad on\ M$$ where $a,b,c\in C^{\infty}(M)$, $a$ and $c$ are positive, $-div_{g}(a\nabla)+b$ is coercive, and $2^{\ast}=\frac{2m}{m-2}$ is the critical Sobolev exponent. In particular, if $R_{g}$ denotes the scalar curvature of $(M,g)$, we give conditions which guarantee that the Yamabe problem $$Δ_{g}u+\frac{m-2}{4(m-1} R_{g}u=κu^{2^{\ast}-2}\quad on\ M$$ admits a prescribed number of nodal solutions.

math.AP↗