Related papers: A variational approach to the quaternionic Hessian…
Several sets of quaternionic functions are described and studied. Residue current of the right inverse of a quaternionic function is introduced in particular cases.
This paper presents an innovative set of tools developed to support a methodology to find the left eigenvalues of $m$ order quaternion square matrix. It is solving four real polynomial equations of order not greater than $4m-3$ in four…
In this paper, we first study the comparison principle for the operator $H_{\chi,m}$. This result is used to solve certain weighted complex $m-$ Hessian equations.
We present a variational formulation of electrodynamics using de Rham even and odd differential forms. Our formulation relies on a variational principle more complete than the Hamilton principle and thus leads to field equations with…
In this paper, we introduce the class $\mathcal{E}_{m,F}(\Omega)$ and solve complex $m$-Hessian equations in the class $\mathcal{E}_{m,F}(\Omega)$. Afterthat, we study subextension in the class $\mathcal{E}_{m,F}(\Omega)$ with the weighted…
This paper is concerned with the following fractional Schr\"odinger equation \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^{s} u+u= k(x)f(u)+h(x) \mbox{ in } \mathbb{R}^{N}\\ u\in H^{s}(\R^{N}), \, u>0 \mbox{ in } \mathbb{R}^{N},…
The aim of this paper is to study Capacities and Hessians in a class of m-subharmonic functions
The goal of this article is to present a survey of the recent theory of plurisubharmonic functions of quaternionic variables, and its applications to theory of valuations on convex sets and HKT-geometry (HyperK\"ahler with Torsion). The…
Using the $H^\infty$-functional calculus for quaternionic operators, we show how to generate the fractional powers of some densely defined differential quaternionic operators of order $m\geq 1$, acting on the right linear quaternionic…
In a recent paper [J.Math.Phys. vol42, 2236-2265 (2001)], we discussed differential operators within a quaternionic formulation of quantum mechanics. In particular, we proposed a practical method to solve quaternionic and complex linear…
In this paper, we introduce the first-order differential operators $d_0$ and $d_1$ acting on the quaternionic version of differential forms on the flat quaternionic space $\mathbb{H}^n$. The behavior of $d_0,d_1$ and $\triangle=d_0d_1$ is…
Due to the non-commutative nature of quaternions we introduce the concept of left and right action for quaternionic numbers. This gives the opportunity to manipulate appropriately the $H$-field. The standard problems arising in the…
The multidimensional Cauchy-Riemann operator provides a framework for studying higher order partial differential equations in $\mathbb{R}^{m+1}$, whose solutions include polymonogenic and polyharmonic functions, among others. In this work,…
On the basis of variational method we study energy levels of pionic helium $(\pi-e-He)$ and kaonic helium $(K-e-He)$ with an electron in ground state and a meson in excited state with principal and orbital quantum numbers $n\sim l+1\sim…
Quaternionic analysis relies heavily on results on functions defined on domains in $\mathbb R^4$ (or $\mathbb R^3$) with values in $\mathbb H$. This theory is centered around the concept of $\psi-$hyperholomorphic functions i.e.,…
We prove a Caffarelli--Kohn--Nirenberg inequality in the hyperbolic space. For a semilinear elliptic equation involving the associated weighted Laplace--Beltrami operator, we establish variationally the existence of positive radial…
Our aim in this article is to study semilinear elliptic equations involving a fractional Hardy operator, an absorption and a Radon source in a weighted distributional sense. We show various scenarios, produced by the combined effect of the…
In this paper we aim to combine tools from variational calculus with modern techniques from quaternionic analysis that involve Dirac type operators and related hypercomplex integral operators. The aim is to develop new methods for showing…
In this paper we firstly introduce the concepts of capacity and Cegrell's classes associated to any $m$-positive closed current $T$. Next, after investigating the most imporant related properties, we study the definition and the continuity…
The method of separation of variables is significant, it has been applied to physics, engineering , chemistry and other fields. It allows to reduce the diffculity of problems by separating the variables from partial differential equation…