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José Carlos Bellido

Publications and source records attributed to José Carlos Bellido.

7 recordsLinked to original sources
arXiv2026

Bessel Potential Spaces and Complex Interpolation: Continuous embeddings

Bessel potential spaces, introduced in the 1960s, are derived through complex interpolation between Lebesgue and Sobolev spaces, making them intermediate spaces of fractional differentiability order. Bessel potential spaces have recently gained attention due to their identification with the Riesz fractional gradient. This paper explores Bessel potential spaces as complex interpolation spaces, providing original proofs of fundamental properties based on abstract interpolation theory. Main results include a direct proof of norm equivalence, continuous embeddings, and the relationship with Gagliardo spaces.

math.FA↗
arXiv2026

Nonlocal gradient, the nonlocal Laplacian and maximum principles

We study the nonlocal $ρ$-Laplacian, defined as the composition of the nonlocal divergence and gradient operators associated with a general radial kernel $ρ$: $Δ_ρu=\mbox{div}_ρ\left(D_ρu\right)$. Our first main contribution is to establish a precise connection between this operator and the class of integro-differential elliptic operators studied by Fernández-Real and Ros-Oton (\cite{FernandezRos}), identifying explicit conditions on the kernel $ρ$ that guarantee membership in this class. Our second main contribution concerns maximum and comparison principles for the $ρ$-Laplacian. We establish both a strong and a weak maximum principle under conditions on $ρ$ that are strictly weaker than those required for membership in the integro-differential class, thereby covering a genuinely broader family of operators. The results require only minimal assumptions on the kernel, and in particular do not rely on any fractional-type comparability condition.

math.AP↗
arXiv2024

Nonlocal gradients: Fundamental theorem of calculus, Poincaré inequalities and embeddings

We address the study of nonlocal gradients defined through general radial kernels $ρ$. Our investigation focuses on the properties of the associated function spaces, which depend on the characteristics of the kernel function. Specifically, even with minimal assumptions on $ρ$, we establish Poincaré inequalities and compact embeddings into Lebesgue spaces. Additionally, we present a fundamental theorem of calculus that enables us to recover a function from its nonlocal gradient through a convolution. This is used to demonstrate embeddings into Orlicz spaces and spaces of continuous functions that mirror the well-known Sobolev and Morrey inequalities for classical gradients. Finally, we establish conditions for inclusions and equality of spaces associated to different kernels.

math.AP↗
arXiv2024

Existence and uniqueness of solutions for Leontief's Input-Output Model, graph theory and sensitivity analysis

We provide a complete study of existence and uniqueness (uniqueness up to multiples in the case $\mathbf{d} = \mathbf{0}$) of non-negative and non-trivial solutions $\mathbf{x}$ for the linear system $(\mathbf{I}- \mathbf{A})\mathbf{x} = \mathbf{d}$ with $\mathbf{A} \geq \mathbf{0}, \mathbf{d} \geq 0$ (which, in particular, applies to Leontief's Input Output Model). This study is done in terms of the block triangular form of the matrix $\mathbf{A}$ and is related to a directed graph associated to $\mathbf{A}$. In particular, this study of existence and uniqueness up to multiples of solutions provides a framework that allows us to rigorously perform a sensitivity analysis of the normalized solutions of the linear system $(\mathbf{I} - \mathbf{A})\mathbf{x} = \mathbf{d}$.

math.OC↗
arXiv2022

Nonlocal gradients in bounded domains motivated by Continuum Mechanics: Fundamental Theorem of Calculus and embeddings

In this paper we develop a new set of results based on a nonlocal gradient jointly inspired by the Riesz s-fractional gradient and Peridynamics, in the sense that its integration domain depends on a ball of radius delta > 0 (horizon of interaction among particles, in the terminology of Peridynamics), while keeping at the same time the singularity of the Riesz potential in its integration kernel. Accordingly, we define a functional space suitable for nonlocal models in Calculus of Variations and partial differential equations. Our motivation is to develop the proper functional analysis framework in order to tackle nonlocal models in Continuum Mechanics, which requires working with bounded domains, while retaining the good mathematical properties of Riesz s-fractional gradients. This functional space is defined consistently with Sobolev and Bessel fractional ones: we consider the closure of smooth functions under the natural norm obtained as the sum of the Lp norms of the function and its nonlocal gradient. Among the results showed in this investigation we highlight a nonlocal version of the Fundamental Theorem of Calculus (namely, a representation formula where a function can be recovered from its nonlocal gradient), which allows us to prove inequalities in the spirit of Poincaré, Morrey, Trudinger and Hardy as well as the corresponding compact embeddings. These results are enough to show the existence of minimizers of general energy functionals under the assumption of convexity. Equilibrium conditions in this nonlocal situation are also established, and those can be viewed as a new class of nonlocal partial differential equations in bounded domains.

math.AP↗
arXiv2016

Worst-case shape optimization for the Dirichlet energy

We consider the optimization problem for a shape cost functional $F(Ω,f)$ which depends on a domain $Ω$ varying in a suitable admissible class and on a "right-hand side" $f$. More precisely, the cost functional $F$ is given by an integral which involves the solution $u$ of an elliptic PDE in $Ω$ with right-hand side $f$; the boundary conditions considered are of the Dirichlet type. When the function $f$ is only known up to some degree of uncertainty, our goal is to obtain the existence of an optimal shape in the worst possible situation. Some numerical simulations are provided, showing the difference in the optimal shape between the case when $f$ is perfectly known and the case when only the worst situation is optimized.

math.OC↗