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Juan Carlos Sampedro

Publications and source records attributed to Juan Carlos Sampedro.

At least 19 recordsLinked to original sources
arXiv2026

Sequential exponential-motivic periods in spatially cut-off $ϕ^4_2$

We represent constructive integrals in spatially cut-off $ϕ^4_2$ by sequences of exponential-motivic periods. For a nonzero spatial cutoff, a single Galois transformation makes the partition periods diverge in modulus along every sequence of approximations satisfying the constructive estimates. For a fixed cutoff presentation, we characterize the subgroup of transformations admitting a bounded realization on the free field's $L^1$ space and prove that all fixed-observable limits on this subgroup are a common scalar multiple of their physical values. A one-variable quartic example shows that all transformed fixed-observable sequences can converge without such a bounded realization, with limits that are not proportional to the physical values.

math.NT↗
arXiv2026

Bifurcation and global continuation of travelling-rotating Schrödinger maps on the sphere

We study travelling-rotating solutions of the Schrödinger map equation into the sphere, viewed as tangent profiles of rigid vortex filaments. Two first integrals reduce the profile equation to a scalar cubic equation for the vertical component, giving an elliptic-function description and explicit closure conditions. We prove bifurcation from the equatorial branch at $λ_k=R\sqrt{k^2-1}$, $k\ge2$, and establish a global continuation alternative inside the regular non-polar class. The possible boundary mechanisms are pole contact, vertical collapse, and double-root degeneration. Numerical continuation of the equatorial branches suggests convergence to the north-pole boundary. Up to gauge, the reconstructed vortex filaments are of Kida type.

math.AP↗
arXiv2026

Note on infinite-dimensional $L^p$-spaces

We prove that, for every $1\leq p<\infty$, the $L^{p}$-space of Baker's measure on $\mathbb{R}^{\mathbb{N}}$ is isometrically isomorphic to $\ell^p(\mathfrak{c},L^{p}[0,1])$ in ZFC. This solves in a negative manner the main problem stated in [Isometric classification of the $L^{p}$-spaces of infinite dimensional Lebesgue measure, Banach J. Math. Anal. 20 (2026), no. 1, Paper No. 7].

math.FA↗
arXiv2026

Distributional embeddings of the first limit Bourgain-Rosenthal-Schechtman space

We classify the distributional self-embeddings of the centered first limit Bourgain-Rosenthal-Schechtman space $R_ω^{p,0}$, $1<p<\infty$. Using a Boolean rigidity principle for its canonical independent-sum realization, we show that every such embedding is induced by a finite packing of Bernoulli factors. As a consequence, we also prove that $R_ω^{p,0}$ admits no proper non-zero internal compressions. Moreover, for $p\notin2\mathbb N$, we obtain a complete description of the linear isometric embeddings of the non-centered space $R_ω^p$, and, for $p\neq2$, we determine its group of surjective linear isometries.

math.FA↗
arXiv2026

A bifurcation theory approach to the nonlocal Kuramoto-Sivashinsky equation

We study the nonlocal Kuramoto-Sivashinsky equation on the one-dimensional torus, \[ u_t+u u_x=Λ^{r}u-\varepsilon Λ^{s}u,\qquad x\in\mathbb T, \] where $\varepsilon>0$, $s>1$, $r\in[-1,s)$. We first prove local and global well-posedness for initial data in $H^{3}(\mathbb T)$. We then investigate the steady-state problem and show that the trivial branch undergoes bifurcation at the critical values $\varepsilon_k=k^{\,r-s}$, $k\in\mathbb N$. Using the Crandall-Rabinowitz theorem we obtain smooth local curves of nontrivial equilibria emanating from each $(\varepsilon_k,0)$ and compute the bifurcation direction. To address the global continuation of these branches we derive global a priori bounds and apply a global alternative based on the Fitzpatrick-Pejsachowicz-Rabier degree for Fredholm maps of index zero. In particular, for the component bifurcating from the first critical point we prove that its $\varepsilon$-projection contains the interval $(2^{r-s},1)$, yielding the existence of nontrivial steady states for that parameter range. We complement the theory with numerical continuation results illustrating the bifurcation diagram and solution profiles.

math.AP↗
arXiv2025

On the $L^{p}$-spaces of projective limits of probability measures

The present article describes the precise structure of the $L^{p}$-spaces of projective limit measures by introducing a category theoretical perspective. This analysis is applied to measures on vector spaces and in particular to Gaussian measures on nuclear topological vector spaces. A simple application to constructive Quantum Field Theory (QFT) is given through the Osterwalder-Schrader axioms.

math.PR↗
arXiv2025

Periodic solutions to nonlocal pseudo-differential equations. A bifurcation theoretical perspective

In this paper we use abstract bifurcation theory for Fredholm operators of index zero to deal with periodic even solutions of the one-dimensional equation $\mathcal{L}u=λu+|u|^{p}$, where $\mathcal{L}$ is a nonlocal pseudodifferential operator defined as a Fourier multiplier and $λ$ is the bifurcation parameter. Our general setting includes the fractional Laplacian $\mathcal{L}\equiv(-Δ)^{s}$ and sharpens the results obtained for this operator to date. As a direct application, we establish the existence of traveling waves for general nonlocal dispersive equations for some velocity ranges.

math.AP↗
arXiv2025

Global Leray-Schauder continuation for Fredholm operators

This paper ascertains the global behavior of the forward and backward branches of solutions provided by the Leray-Schauder continuation theorem for orientable $\mathcal{C}^1$ Fredholm maps, as developed by the authors in [54]. Under properness on bounded sets and a nonzero local index at the given base solution, each branch satisfies the following alternative: either it is unbounded, or it reaches the boundary of the domain, or it accumulates at a different solution on the base parameter level. When the component is bounded and stays in the interior, there is a degree balance on the base slice entailing a vanishing sum of local indices and, in particular, the existence of an even number of non-degenerate contact points. For real-analytic maps we construct locally injective parameterizations that exhibit blow-up, approach to the boundary, or return to the base level. An application to a quasilinear boundary value problem driven by the mean-curvature and Minkowski operators illustrates the global results.

math.AP↗
arXiv2025

Isometric classification of the $L^{p}$-spaces of infinite dimensional Lebesgue measure

We investigate the isometric structure of $L^{p}$-spaces for the infinite-dimensional Lebesgue measure $(\mathbb{R}^{\mathbb{N}},μ)$. Under the continuum hypothesis (CH) we prove $L^{p}(μ)\cong \ell^{p}(\mathfrak{c},L^{p}[0,1])$, where $\mathfrak{c}$ denotes the cardinality of the continuum, and without CH we obtain an isometric, complemented copy of $\ell^{p}(\mathfrak{c},L^{p}[0,1])$ inside $L^{p}(μ)$. In a general framework, we characterize precisely when $L^{p}(ν)\cong \ell^{p}(κ,L^{p}[0,1])$ and classify all such isometries.

math.FA↗
arXiv2025

Spectral Theory and Bezout Theorem

This note investigates the hidden relationship between the concept of algebraic multiplicity of an eigenvalue and the local intersection index of algebraic varieties.

math.SP↗
arXiv2025

Bifurcation Theory for a Class of Periodic Superlinear Problems

We analyze, mainly using bifurcation methods, an elliptic superlinear problem in one-dimension with periodic boundary conditions. One of the main novelties is that we follow for the first time a bifurcation approach, relying on a Lyapunov-Schmidt reduction and some recent global bifurcation results, that allows us to study the local and global structure of non-trivial solutions at bifurcation points where the linearized operator has a two-dimensional kernel. Indeed, at such points the classical tools in bifurcation theory, like the Crandall-Rabinowitz theorem or some generalizations of it, cannot be applied because the multiplicity of the eigenvalues is not odd, and a new approach is required. We apply this analysis to specific examples, obtaining new existence and multiplicity results for the considered periodic problems, going beyond the information variational and fixed point methods like Poincaré-Birkhoff theorem can provide.

math.CA↗
arXiv2024

Existence of Infinite Product Measures

A construction of product measures is given for an arbitrary sequence of measure spaces via outer measure techniques without imposing any condition on the underlying measure spaces. This result generalises the ones given up to date.

math.FA↗
arXiv2024

Existence of Infinite Product Measures

A construction of product measures is given for an arbitrary sequence of measure spaces via outer measure techniques without imposing any condition on the underlying measure spaces. This approach concludes finally the problem of the existence of product measures in an elementary manner. Moreover, the $L_{p}$ spaces of this measures are simplified in terms of finite product measures following the approach of [21]. This decomposition simplifies infinite dimensional integration and gives to this theory a computational framework.

math.FA↗
arXiv2024

Blow-up estimates and a priori bounds for the positive solutions of a class of superlinear indefinite elliptic problems

In this paper we find out some new blow-up estimates for the positive explosive solutions of a paradigmatic class of elliptic boundary value problems of superlinear indefinite type. These estimates are obtained by combining the scaling technique of Guidas-Spruck together with a generalized De Giorgi-Moser weak Harnack inequality found, very recently, by Sirakov. In a further step, based on a comparison result of Amann and López-Gómez, we will show how these bounds provide us with some sharp a priori estimates for the classical positive solutions of a wide variety of superlinear indefinite problems. It turns out that this is the first general result where the decay rates of the potential in front of the nonlinearity $a(x)$ do not play any role for getting a priori bounds for the positive solutions when $N\geq 3$.

math.AP↗
arXiv2023

Bifurcation Theory for Fredholm Operators

This paper consists of four parts. It begins by using the authors's generalized Schauder formula, [50], and the algebraic multiplicity, $χ$, of Esquinas and López-Gómez [18,17,40] to package and sharpening all existing results in local and global bifurcation theory for Fredholm operators through the recent author's axiomatization of the Fitzpatrick-Pejsachowicz-Rabier degree, [51]. This facilitates reformulating and refining all existing results in a compact and unifying way. Then, the local structure of the solution set of analytic nonlinearities $\mathfrak{F}(λ,u)=0$ at a simple degenerate eigenvalue is ascertained by means of some concepts and devices of Algebraic Geometry and Galois Theory, which establishes a bisociation between Bifurcation Theory and Algebraic Geometry. Finally, the unilateral theorems of [40,42], as well as the refinement of Xi and Wang [63], are substantially generalized. This paper also analyzes two important examples to illustrate and discuss the relevance of the abstract theory. The second one studies the regular positive solutions of a multidimensional quasilinear boundary value problem of mixed type related to the mean curvature operator.

math.AP↗
arXiv2022

Periodic solutions to superlinear indefinite planar systems: a topological degree approach

We deal with a planar differential system of the form \begin{equation*} \begin{cases} \, u' = h(t,v), \\ \, v' = - λa(t) g(u), \end{cases} \end{equation*} where $h$ is $T$-periodic in the first variable and strictly increasing in the second variable, $λ>0$, $a$ is a sign-changing $T$-periodic weight function and $g$ is superlinear. Based on the coincidence degree theory, in dependence of $λ$, we prove the existence of $T$-periodic solutions $(u,v)$ such that $u(t)>0$ for all $t\in\mathbb{R}$. Our results generalize and unify previous contributions about Butler's problem on positive periodic solutions for second-order differential equations (involving linear or $ϕ$-Laplacian-type differential operators).

math.CA↗
arXiv2022

Generating loops and isolas in semilinear elliptic BVP's

In this paper, we ascertain the global $λ$-structure of the set of positive and negative solutions bifurcating from $u=0$ for the semilinear elliptic BVP \begin{equation*} \left\{\begin{array}{ll} -dΔu= λ\langle \mathfrak{a},\nabla u\rangle+u+λu^{2}-u^{q} & \text{ in } Ω, \\ u=0 & \text{ on } \partialΩ, \end{array}\right. \end{equation*} according to the values of $d>0$ and the integer number $q\geq 4$. Figures 1.1-1.3 summarize the main findings of this paper according to the values of $d$ and $q$. Note that the role played by the parameter $λ$ in this model is very special, because, besides measuring the strength of the convection, it quantifies the amplitude of the nonlinear term $λu^2$. We regard to this problem as a mathematical toy to generate solution loops and isolas in Reaction Diffusion equations.

math.AP↗