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Briceyda B. Delgado

Publications and source records attributed to Briceyda B. Delgado.

10 recordsLinked to original sources
arXiv2026

Transmutation operators for the Sturm--Liouville equation in impedance form with an integrable potential

In this work, we study a family of transmutation operators for the Sturm--Liouville operator in impedance form with an integrable potential satisfying a smallness condition. Specifically, we consider transmutations associated with the pair \[\textbf{L}_r=\frac{1}{r^2 (x)}\frac{d}{dx}r^2 (x)\frac{d}{dx} \quad \text{and} \quad \textbf{M}=\frac{d^2}{dx^2}. \] We extend existing results by showing that a particular transmutation operator on $W^{3,1} (-a,a)$ can be represented in integral form, where the kernel function satisfies a corresponding Goursat problem. We establish the existence of this kernel using two distinct approaches: the classical method of successive approximations and an application of Picard's theorem.

math.SP↗
arXiv2026

A biquaternionic reformulation of Maxwell's equations via Fourier analysis

We analyze the parabolic Dirac operator $D \pm i\partial_t$ in a biquaternionic setting, characterizing its kernel via generalized div-curl systems and Cauchy-Riemann-type relations between the real and imaginary parts. Using the machinery provided by the Fourier transform, we fully characterize the Fourier transform of a fundamental solution of this operator and construct a well-defined right inverse operator. As an application, we derive explicit vectorial solutions to the time-dependent Maxwell system, extending prior biquaternionic approaches. These tools offer analytical efficiency for complex electromagnetic problems.

math.AP↗
arXiv2026

Hodge decomposition for generalized Vekua spaces in higher dimensions

We introduce the spaces $A^p_{α, β}(Ω)$ of $L^p$-solutions to the Vekua equation (generalized monogenic functions) $D w=α\overline{w}+βw$ in a bounded domain in $\mathbb{R}^n$, where $D=\sum_{i=1}^n e_i \partial_i$ is the Moisil-Teodorescu operator, $α$ and $β$ are bounded functions on $Ω$. The main result of this work consists of a Hodge decomposition of the $L^2$ solutions of the Vekua equation, from this orthogonal decomposition arises an operator associated with the Vekua operator, which in turn factorizes certain Schrödinger operators. Moreover, we provide an explicit expression of the ortho-projection over $A^p_{α, β}(Ω)$ in terms of the well-known ortho-projection of $L^2$ monogenic functions and an isomorphism operator. Finally, we prove the existence of component-wise reproductive Vekua kernels and the interrelationship with the Vekua projection in Bergman's sense.

math.AP↗
arXiv2026

Calderón's inverse problem via Vekua theory

In this work, we will prove a uniqueness result for Calderón's inverse problem via some integral representation formulas for solutions of the Vekua equation in the framework of Clifford analysis.

math.AP↗
arXiv2023

A right inverse of curl operator which is divergence free invariant and some applications to generalized Vekua type problems

In this work, we investigate the system formed by the equations $\text{div } \vec w=g_0$ and $\text{curl } \vec w=\vec g$ in bounded star-shaped domains of $\mathbb{R}^3$. A Helmholtz-type decomposition theorem is established based on a general solution of the above-mentioned div-curl system which was previously derived in the literature. When $g_0\equiv 0$, we readily obtain a bounded right inverse of $\text{curl}$ which is a divergence-free invariant. The restriction of this operator to the subspace of divergence-free vector fields with vanishing normal trace is the well-known Biot--Savart operator. In turn, this right inverse of $\text{curl}$ will be modified to guarantee its compactness and satisfy suitable boundary-value problems. Applications to Beltrami fields, Vekua-type problems as well as Maxwell's equations in inhomogeneous media are included.

math-ph↗
arXiv2021

Bifurcation and Multiplicity Results for Elliptic Problems with Subcritical Nonlinearity on the Boundary

We consider an elliptic problem with nonlinear boundary condition involving nonlinearity with superlinear and subcritical growth at infinity and a bifurcation parameter as a factor. We use re-scaling method, degree theory and continuation theorem to prove that there exists a connected branch of positive solutions bifurcating from infinity when the parameter goes to zero. Moreover, if the nonlinearity satisfies additional conditions near zero, we establish a global bifurcation result, and discuss the number of positive solution(s) with respect to the parameter using bifurcation theory and degree theory.

math.AP↗
arXiv2018

A right inverse operator for $\operatorname{curl}+λ$ and applications

A general solution of the equation $\operatorname{curl}\vec{w}+λ\vec {w}=\overrightarrow{g},\,λ\in\mathbb{C},\,λ\neq0$ is obtained for an arbitrary bounded domain $Ω\subset\mathbb{R}^{3}$ with a Liapunov boundary and $\overrightarrow{g}\in W^{p,\operatorname{div}}\left( Ω\right) =\left\{ \overrightarrow{u}\in L^{p}\left( Ω\right) :\,\operatorname{div}\overrightarrow{u}\in L^{p}\left( Ω\right) ,\,1<p<\infty\right\} $. The result is based on the use of classical integral operators of quaternionic analysis. Applications of the main result are considered to a Neumann boundary value problem for the equation $\operatorname{curl}\vec{w}+λ\vec {w}=\overrightarrow{g}$ as well as to the nonhomogeneous time-harmonic Maxwell system for achiral and chiral media.

math-ph↗
arXiv2018

Hilbert transform for the three-dimensional Vekua equation

The three-dimensional Hilbert transform takes scalar data on the boundary of a domain in R3 and produces the boundary value of the vector part of a quaternionic monogenic (hyperholomorphic) function of three real variables, for which the scalar part coincides with the original data. This is analogous to the question of the boundary correspondence of harmonic conjugates. Generalizing a representation of the Hilbert transform H in R3 given by T. Qian and Y. Yang (valid in Rn), we define the Hilbert transform Hf associated to the main Vekua equation DW = (Df/f)W in bounded Lipschitz domains in R3. This leads to an investigation of the three-dimensional analogue of the Dirichlet-to-Neumann map for the conductivity equation.

math.AP↗
arXiv2017

General solution of the inhomogenous div-curl system and consequences

We consider the inhomogeneous div-curl system (i.e.\ to find a vector field with prescribed div and curl) in a bounded star-shaped domain in $\mathbb{R}^3$. An explicit general solution is given in terms of classical integral operators, completing previously known results obtained under restrictive conditions. This solution allows us to solve questions related to the quaternionic main Vekua equation $DW=(Df/f)\overline W$ in $\mathbb{R}^3$, such as finding the vector part when the scalar part is known. In addition, using the general solution to the div-curl system and the known existence of the solution of the inhomogeneous conductivity equation, we prove the existence of solutions of the inhomogeneous double curl equation, and give an explicit solution for the case of static Maxwell's equations with only variable permeability.

math-ph↗