arXiv subjects

María de los Ángeles Sandoval-Romero

Publications and source records attributed to María de los Ángeles Sandoval-Romero.

6 recordsLinked to original sources

arXiv2026

Global Analysis: An Introduction to Nonlinear Analysis and Its Variational Methods on Riemannian Manifolds

This monograph develops an introduction to global analysis centered on the interaction between differential geometry, functional analysis, partial differential equations, and variational methods on Riemannian manifolds. Beginning with smooth and Riemannian geometry, it develops Sobolev spaces, distributions, interpolation and fractional regularity, differential and pseudodifferential operators on vector bundles, elliptic theory, heat methods, bounded geometry, and trace theorems. It then treats Fredholm and index theory, culminating in the Atiyah--Singer index theorem, followed by geometric evolution equations and Ricci flow, infinite-dimensional geometry on Banach and Hilbert manifolds, and variational methods including the direct method, Palais--Smale theory, deformation arguments, the mountain pass theorem, and the Nehari method. Particular emphasis is placed on explicit proofs, the passage from local Euclidean estimates to intrinsic global statements, and the precise geometric hypotheses required in compact, noncompact, and boundary settings. The text is intended for advanced undergraduate and graduate students, as well as readers approaching global analysis from geometry or differential equations.

math.DG↗

arXiv2026

An Intrinsic Sobolev Trace Theory for Riemannian Vector Bundles with Applications to Quasilinear Elliptic Equations

Let $\mathbf{E}\to M$ be a finite-rank Riemannian vector bundle over a compact Riemannian manifold with nonempty smooth boundary, equipped with a compatible connection. We develop an intrinsic trace theory for Sobolev sections defined through weak covariant derivatives. For every integer $m\ge1$ and every $1\le p<\infty$, we construct continuous trace operators for the covariant derivatives up to order $m-1$ and characterize the Sobolev space with homogeneous Dirichlet boundary conditions as the kernel of the corresponding trace operator. As an application, we establish the existence of ground state solutions for a class of quasilinear elliptic equations on Riemannian vector bundles with generalized $(p,q)$-growth. The proposed framework unifies several important geometric operators, including the Bochner $p$-Laplacian, the $(p,q)$-Bochner Laplacian, and prescribed mean curvature-type operators.

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

Introduction to Measure and Integration Theory

These notes provide a rigorous and accessible introduction to measure and integration theory, with emphasis on the conceptual transition from the Riemann integral to the Lebesgue integral and the role played by limiting processes in modern analysis. The manuscript develops the basic theory of measurable sets, measurable functions, measures on $σ$-algebras, Lebesgue integration, convergence theorems, and $L^p$ spaces. Particular attention is devoted to the interaction between integration and convergence, as well as to the limitations of the Riemann integral that motivate the development of measure theory. The exposition seeks to balance mathematical rigor with pedagogical clarity through detailed proofs, examples, exercises, and supplementary projects. These notes are intended primarily for undergraduate students in mathematics and related areas encountering measure theory for the first time, although they may also serve as a reference for introductory graduate courses in analysis.

math.HO↗

arXiv2025

An Analytic Construction of Random Variables in Lebesgue Spaces

This work develops, from a functional analytic perspective, the construction of random variables in Lebesgue spaces L^p. It extends classical notions of measurability, integrability, and expectation to L^p valued functions, using Pettis's theorem and the Riesz representation theorem to define the Bochner integral as a natural generalization of classical expectation.

math.PR↗

arXiv2025

Variational proof of conditional expectations

In this paper, we show that the conditional expectation of a random variable with finite second moment given a $σ$-algebra is the unique critical point of an energy functional in Hilbert space $L^2$. Then, we extend by density the result to every integrable random variable.

math.PR↗