arXiv subjects

J. A. Barceló

Publications and source records attributed to J. A. Barceló.

4 recordsLinked to original sources

arXiv2026

A new addition theorem for the 3-D Navier-Lamé system and its application to the method of fundamental solutions

We obtain a new addition theorem for the fundamental solution of the Navier-Lamé system in dimension 3 satisfying the Kupradze radiation conditions. This provides an expansion of this fundamental solution that involves only the evaluation of Bessel functions and scalar spherical harmonics. This is particularly useful in collocation numerical methods based on fundamental solutions, such as the boundary element method or the method of fundamental solutions. For this last method, we show its efficiency when approximating the Navier-Lamé system in exterior domains.

math.NA↗

arXiv2022

Live load matrix recovery from scattering data in linear elasticity

We study the numerical approximation of the inverse scattering problem in the two-dimensional homogeneous isotropic linear elasticity with an unknown linear load given by a square matrix. For both backscattering data and fixed-angle scattering data, we show how to obtain numerical approximations of the so-called Born approximations and propose new iterative algorithms that provide sequences of approximations to the unknown load. Numerical evidences of the convergence for not too large loads are also given.

math.AP↗

arXiv2020

The Fourier extension operator of distributions in Sobolev spaces of the sphere and the Helmholtz equation

The purpose of this paper is to characterize all the entire solutions of the homogeneous Helmholtz equation (solutions in $\mathbb{R}^d$) arising from the Fourier extension operator of distributions in Sobolev spaces of the sphere $H^α(\mathbb{S}^{d-1}),$ with $α\in \mathbb{R}$. We present two characterizations. The first one is written in terms of certain $L^2$-weighted norms involving real powers of the spherical Laplacian. The second one is in the spirit of the classical description of the Herglotz wave functions given by P. Hartman and C. Wilcox. For $α>0$ this characterization involves a multivariable square function evaluated in a vector of entire solutions of the Helmholtz equation, while for $α<0$ it is written in terms of an spherical integral operator acting as a fractional integration operator. Finally, we also characterize all the solutions that are the Fourier extension operator of distributions in the sphere.

math.CA↗

arXiv2019

Characterization of Sobolev spaces on the sphere

We prove a characterization of the Sobolev spaces $H^α$ on the unit sphere $\mathbb{S}^{d-1}$, where the smoothness index $α$ is any positive real number and $d\geq 2$. This characterization does not use differentiation and it is given in terms of $([α/2]+1)$-multidimensional square functions $S_α$. For $[α/2]=0,$ a function $f\in L^2(\mathbb{S}^{d-1})$ belongs to $H^α(\mathbb{S}^{d-1})$ if and only if $S_α(f)\in L^2(\mathbb{S}^{d-1})$. If $n=[α/2]>0$, the membership of $f$ is equivalent to the existence of $g_1,\cdots,g_n$ in $L^2(\mathbb{S}^{d-1})$ such that $S_α(f,g_1,\ldots,g_n)\in L^2(\mathbb{S}^{d-1})$ and in this case, $g_j=T_j((-Δ_S)^j f)$, where $T_j$ is a zonal Fourier multiplier in the sphere and $Δ_S$ is the Laplace-Beltrami operator. The square functions $S_α$ are based on averaging operators over euclidean balls (caps) in the sphere that may be viewed as zonal multipliers. The results in the paper are in the spirit of the characterization of fractional Sobolev spaces given in $\mathbb{R}^d$ proved in \cite{AMV}. The development of the theory is fully based on zonal Fourier multipliers and special functions.

math.CA↗