arXiv subjects

Alejandro Ortega

Publications and source records attributed to Alejandro Ortega.

30 records · Page 2Linked to original sources

arXiv2021

Bound and ground states of coupled "NLS-KDV" equations with Hardy potential and critical power

We consider the existence of bound and ground states for a family of nonlinear elliptic systems in $\mathbb{R}^N$, which involves equations with critical power nonlinearities and Hardy-type singular potentials. The equations are coupled by what we call ``Schrödinger-Korteweg-de Vries'' non-symmetric terms, which arise in some phenomena of fluid mechanics. By means of variational methods, ground states are derived for several ranges of the positive coupling parameter $ν$. Moreover, by using min-max arguments, we seek bound states under some energy assumptions.

math.AP↗

arXiv2021

Existence of bound and ground states for an elliptic system with double criticality

We study the existence of bound and ground states for a class of nonlinear elliptic systems in $\mathbb{R}^N$. These equations involve critical power nonlinearities and Hardy-type singular potentials, coupled by a term containing up to critical powers. More precisely, we find ground states either the positive coupling parameter $ν$ is large or $ν$ is small under suitable assumptions on the other parameters of the problem. Furthermore, bound states are found as Mountain-Pass-type critical points of the underlying functional constrained on the Nehari manifold. Our variational approach improves some known results and allows us to cover range of parameters which have not been considered previously.

math.AP↗

arXiv2021

On the motion of gravity-capillary waves with odd viscosity

We develop three asymptotic models of surface waves in a non-newtonian fluid with odd viscosity. This viscosity is also known as Hall viscosity and appears in a number of applications such as quantum Hall fluids or chiral active fluids. Besides the odd viscosity effects, these models capture both gravity and capillary forces up to quadratic interactions and take the form of nonlinear and nonlocal wave equations. Two of these models describe bidirectional waves while the third PDE studies the case of unidirectional propagation. We also prove the well-posedness of these asymptotic models in spaces of analytic functions and in Sobolev spaces. Finally, we present a number of numerical simulations for the unidirectional model.

math.AP↗

arXiv2021

On the Robin function for the Fractional Laplacian on symmetric domains

In this work we prove the non-degeneracy of the critical points of the Robin function for the Fractional Laplacian under symmetry and convexity assumptions on the domain $Ω$. This work extends to the fractional setting the results of M. Grossi concerning the classical Laplace operator.

math.AP↗

arXiv2020

Spectral stability for the perydinamic fractional $p$-Laplacian

In this work we analyze the behavior of the spectrum of the peridynamic fractional $p$-Laplacian, $(-Δ_p)_δ^s$, under the limit process $δ\to0^+$ or $δ\to+\infty$. We prove spectral convergence to the classical $p$-Laplacian under a suitable scaling as $δ\to0^+$ and to the fractional $p$-Laplacian as $δ\to+\infty$.

math.AP↗

arXiv2020

A restricted nonlocal operator bridging together the Laplacian and the Fractional Laplacian

In this work we introduce volume constraint problems involving the nonlocal operator $(-Δ)_δ^{s}$, closely related to the fractional Laplacian $(-Δ)^{s}$, and depending upon a parameter $δ>0$ called horizon. We study the associated linear and spectral problems and the behavior of these volume constraint problems when $δ\to0^+$ and $δ\to+\infty$. Through these limit processes on $(-Δ)_δ^{s}$ we derive spectral convergence to the local Laplacian and to the fractional Laplacian as $δ\to 0^+$ and $δ\to +\infty$ respectively, as well as we prove the convergence of solutions of these problems to solutions of a local Dirichlet problem involving $(-Δ)$ as $δ\to0^+$ or to solutions of a nonlocal fractional Dirichlet problem involving $(-Δ)^s$ as $δ\to+\infty$.

math.AP↗

arXiv2019

Positive solutions for semilinear fractional elliptic problems involving an inverse fractional operator

This paper is devoted to the study of the existence of positive solutions for a problem related to a higher order fractional differential equation involving a nonlinear term depending on a fractional differential operator, $$(-Δ)^α u=λu+ (-Δ)^β|u|^{p-1}u \quad \mbox{in}\quad Ω;\qquad (-Δ)^{j}u=0\quad \mbox{on}\quad \partialΩ,\quad \mbox{for}\quad j\in\mathbb{Z},\: 0\leq j< [α]$$ where $Ω$ is a bounded domain in $\mathbb{R}^{N}$, $0<β<1$, $β<α<β+1$ and $λ>0$. In particular, we study the fractional elliptic problem, $$ (-Δ)^{α-β} u= λ(-Δ)^{-β}u+ |u|^{p-1}u \quad\mbox{in} \quad Ω;\qquad u=0 \quad \hbox{on} \quad \partialΩ,$$ and we prove existence or nonexistence of positive solutions depending on the parameter $λ>0$, up to the critical value of the exponent $p$, i.e., for $1<p\leq 2_μ^*-1$ where $μ:=α-β$ and $2_μ^*=\frac{2N}{N-2μ}$ is the critical exponent of the Sobolev embedding.

math.AP↗

arXiv2019

Homotopy regularization for a high-order parabolic equation

In this work we study the solvability of the Cauchy Problem for a quasilinear degenerate high-order parabolic equation \begin{equation*} \left\{ \begin{tabular}{lcl} $u_t=(-1)^{m-1}\nabla\cdot(f^n(|u|)\nablaΔ^{m-1}u)$ & &in $\mathbb{R}^N\times\mathbb{R}_+$, $u(x,0)=u_0(x)$& & in $\mathbb{R}^N$, \end{tabular} \right. \end{equation*} with $m\in\mathbb{N},\ m>1$ and $n>0$ a fixed exponent. Moreover, $f$ is a continuous monotone increasing positive bounded function with $f(0)=0$ and the initial data $u_0(x)$ is bounded smooth and compactly supported. Thus, through an homotopy argument based on an analytic $\varepsilon$-regularization of the degenerate term $f^n(|u|)$ we are able to extract information about the solutions inherited from the polyharmonic equation when $n=0$.

math.AP↗

arXiv2019

Existence of positive solutions for a Brezis--Nirenberg type problem involving an inverse operator

This paper is devoted to the existence of positive solutions for a problem related to a fourth-order differential equation involving a nonlinear term depending on a second order differential operator, $$(-Δ)^2 u=λu+ (-Δ)|u|^{p-1}u,$$ in a bounded domain $Ω\subset\mathbb{R}^N$, $N\geq 7$, and assuming homogeneous Navier boundary conditions. In particular, we study a second order equation involving a nonlocal term of the form, $$-Δu=λ(-Δ)^{-1} u+|u|^{p-1}u,$$ under Dirichlet boundary conditions and we prove the existence of positive solutions depending on the positive real parameter $λ>0$, up to the critical value of the exponent $p$, i.e., when $1<p\leq 2^*-1$, where $2^*=\frac{2N}{N-2}$ is the critical Sobolev exponent. For $p=2^*-1$, this equivalence leads us to a Brezis--Nirenberg type problem, cf. \cite{BN}, but, in our particular case, the linear term is a nonlocal term. The effect that this nonlocal term has on the equation changes the dimensions for which the classical technique based on the minimizers of the Sobolev constant ensures the existence of solution, going from dimensions $N\geq 4$ in the classical Brezis-Nirenberg problem, to dimensions $N\geq7$ for this nonlocal problem.

math.AP↗

arXiv2018

The Brezis-Nirenberg problem for the fractional Laplacian with mixed Dirichlet-Neumann boundary conditions

In this work we study the existence of solutions to the critical Brezis-Nirenberg problem when one deals with the spectral fractional Laplace operator and mixed Dirichlet-Neumann boundary conditions, i.e., $$ \left\{\begin{array}{rcl} (-Δ)^su & = & λu+u^{2_s^*-1},\quad u>0\quad\mbox{in}\quad Ω,\\ u & = & 0\quad\mbox{on}\quad Σ_{\mathcal{D}},\\ \displaystyle\frac{\partial u}{\partial ν} & = & 0\quad\mbox{on}\quad Σ_{\mathcal{N}}, \end{array}\right. $$ where $Ω\subset\mathbb{R}^N$ is a regular bounded domain, $\frac{1}{2}<s<1$, $2_s^*$ is the critical fractional Sobolev exponent, $0\leλ\in \mathbb{R}$, $ν$ is the outwards normal to $\partialΩ$, $Σ_{\mathcal{D}}$, $Σ_{\mathcal{N}}$ are smooth $(N-1)$-dimensional submanifolds of $\partialΩ$ such that $Σ_{\mathcal{D}}\cupΣ_{\mathcal{N}}=\partialΩ$, $Σ_{\mathcal{D}}\capΣ_{\mathcal{N}}=\emptyset$, and $Σ_{\mathcal{D}}\cap\overlineΣ_{\mathcal{N}}=Γ$ is a smooth $(N-2)$-dimensional submanifold of $\partialΩ$.

math.AP↗

arXiv2015

On the effect of boundaries in two-phase porous flow

In this paper we study a model of an interface between two fluids in a porous medium. For this model we prove several local and global well-posedness results and study some of its qualitative properties. We also provide numerics.

math.AP↗

arXiv2014

Study of Structural Properties in Complex Fluids by Addition of Surfactants Using DPD Simulation

In this work we study the tertiary structure of ionic surfactants when the pH in the system is modified using electrostatic dissipative particle dynamics simulations (DPD). The dependence with pH and kind of surfactant is presented. Our simulations reproduce the experimental behavior reported in the literature. The scaling for the radius of gyration with the size of the molecule as a function of pH is also obtained.

physics.chem-ph↗