arXiv subjects

Ignacio Ceresa Dussel

Publications and source records attributed to Ignacio Ceresa Dussel.

7 recordsLinked to original sources

arXiv2026

An eigenvalue problem for a generalized polyharmonic operator in Orlicz-Sobolev spaces without the $Δ_2$-condition

In this paper, we consider a generalized polyharmonic eigenvalue problem of the form $A(u)= λh(u)$ in a bounded smooth domain with Dirichlet boundary conditions in the setting of higher-order Orlicz-Sobolev spaces. Here, $A$ is a very general operator depending on $u$ and arbitrary higher-order derivatives of $u$, whose growth is governed by an Orlicz function, and $h$ is a lower order term. Combining the theories of pseudomonotone operators with complementary systems, we prove that this eigenvalue problem has an infinite number of eigenfunctions and that the corresponding sequence of eigenvalues tends to infinite. We point out that the $Δ_2$-condition is not assumed for the involved Orlicz functions.

math.AP↗

arXiv2026

Optimal Control Strategies for Epidemic Dynamics: Integrating SIR-SI and Lotka--Volterra Models

In this work we present a mathematical model that integrates the epidemiological dynamics of a vector-borne disease (SIR-SI) with Lotka Volterra predator prey ecological interactions. The study analyzes how the presence of natural predators acts as a biological control mechanism to regulate the vector population and, consequently, disease transmission in host. We introduce the concept of the ecological reproduction number, a threshold that links the amplitude of predator prey cycles to disease persistence, showing that natural control depends critically on the ratio between the maximum vector density and the minimum predator density. In scenarios where natural control is insufficient, we formulate an optimal control problem based on the release of predators. Using the Pontryagin Maximum Principle, we characterize the optimal strategy that minimizes the cumulative number of infected individuals and intervention costs, while simultaneously maximizing the susceptible host population at the end of the time horizon. Numerical simulations validate the effectiveness of the model, showing that external intervention mitigates the epidemic peak and stabilizes the system against the natural oscillations of biological populations.

math.OC↗

arXiv2025

$Γ-$convergence of energy functionals in fractional Orlicz spaces beyond the $Δ_2$ condition

Given a Young function $A$, $n\geq 1$ and $s\in(0,1)$ we consider the energy functional $$ \mathcal{J}_s(u)=(1-s)\iint_{\mathbb{R}^n\times \mathbb{R}^n} A\left(\frac{|u(x)-u(y)|}{|x-y|^s}\right)\frac{dxdy}{|x-y|^n}. $$ Without assuming the $Δ_2$ condition on $A$ not its conjugated function $\bar A$, we prove the following liminf inequality: if $u\in E^A(\mathbb{R}^n)$ and $\{u_k\}_{k\in\mathbb{N}}\subset E^A(\mathbb{R}^n)$ is such that $u_k\to u$ in $E^A(\mathbb{R}^n)$, and $s_k\to 1$, then $$ \mathcal{J}(u) \leq \liminf_{k\to\infty } \mathcal{J}_{s_k}(u_k), $$ where $\mathcal{J}$ is a limit functional related with the behavior of the fractional Orlicz-Sobolev spaces as $s\to 1^+$. As a direct consequence, we obtain the $Γ-$convergence of the functional $\mathcal{J}_s$. Finally, we extend our result to the study of the so called \emph{fractional peridynamic} case.

math.AP↗

arXiv2025

Fractional Lane-Emden Hamiltonian systems

In this work, our interest lies in proving the existence of solutions to the following Fractional Lane-Emden Hamiltonian system: $$ \begin{cases} (-Δ)^s u = H_v(x,u,v) & \text{in }Ω,\\ (-Δ)^s v = H_u(x,u,v) & \text{in }Ω,\\ u=v=0 & \text{in } \R^n\setminusΩ. \end{cases} $$ The method, that can be traced back to the work of De Figueiredo and Felmer \cite{DF-F}, is flexible enough to deal with more general nonlocal operators and make use of a combination of fractional order Sobolev spaces together with functional calculus for self-adjoint operators.

math.AP↗

arXiv2024

Peridynamics and Anisotropic Fractional Sobolev Spaces with Variable Exponents

In this paper, our primary objective is to develop the peridynamic fractional Sobolev space and establish novel BBM-type results associated with it. We also address the peridynamic fractional anisotropic $p-$Laplacian. A secondary objective is to explore anisotropic fractional Sobolev spaces with variable exponents, where we also derive new BBM-type results. Additionally, we address the eigenvalue problem in the isotropic case.

math.AP↗

arXiv2024

Shape optimization problems involving nonlocal and nonlinear operators

In this research, we investigate a general shape optimization problem in which the state equation is expressed using a nonlocal and nonlinear operator. We prove the existence of a minimum point for a functional $F$ defined on the family of all 'quasi-open' subsets of a bounded open set $Ω$ in $\mathbb{R}^n$. This is ensured under the condition that $F$ demonstrates decreasing behavior concerning set inclusion and is lower semicontinuous with respect to a suitable topology associated with the fractional $p$-Laplacian under Dirichlet boundary conditions. Moreover, we study the asymptotic behavior of the solutions when $s\to1$ and extend this result to the anisotropic case.

math.AP↗

arXiv2022

A Bourgain-Brezis-Mironescu formula for anisotropic fractional Sobolev spaces and applications to anisotropic fractional differential equations

In this paper we prove Bourgain-Brezis-Mironescu's type results (cf. \cite{BBM2001}) (BBM for short) for an energy functional which is strongly related to the fractional anisotropic p-Laplacian. We also provide with the analogous of Maz'ya-Shaposhnikova (see \cite{MS}) type results for these energies and finally we apply these results to analyze the stability of solutions to anisotropic fractional $p-$laplacian equations.

math.AP↗