The global Borel-Pompeiu-type and Cauchy-type integral formulas are proposed by examining the connection between the space of generalized partial-slice regular functions and a non-constant coefficient differential operator specified by G=& Vert;x ->& Vert;2 & sum;k=1p partial derivative xk+x ->& sum;k=p+1p+qxk partial derivative xk . Structural sets, psi -differential operators, and reproducing functions like A psi and B psi to formulate formulas are introduced. These formulas generalize existing results for quaternionic and Clifford slice monogenic functions, enabling new proofs of properties of generalized partial-slice monogenic functions, thus enriching Clifford analysis theory with potential applications in geometry, physics, and engineering.
In this paper we investigate the solutions of Cauchy problem for a class of three and four-dimensional degenerate partial differential equations of second order with singular coefficients in the complex domain.
In this paper we are going to show the existence of nonnegative ground state solution for a class of system involving a quasilinear elliptic operator and a nonlinearity associated which was studied considering three types of growth: subcritical, critical and supercritical. Such class of operators includes the Laplacian, p-Laplacian and p&q -Laplacian operators.
In this paper, we first introduce poly generalized partial-slice monogenic functions, which include poly-slice monogenic functions and generalized partial-slice monogenic functions as special cases. Subsequently, we present some of their properties, such as the splitting lemma, the decomposition theorem, the generalized representation formula, and three types of Cauchy integral formulas. Finally, we introduce a new type of functions which are closely related to the generalized Vekua-type problem, and establish the relation between this type of functions and poly generalized partial-slice monogenic functions.
This paper investigates the asymptotic behaviour of a specific class of orthogonal trigonometric polynomials (OTPs) on the real axis. While these OTPs can, in principle, be studied via a mapping to orthogonal polynomials on the unit circle (OPUC), a direct asymptotic analysis within the periodic setting remains notably underdeveloped. To address this gap, we develop a direct approach by formulating and solving a 2 & times;2 matrix-valued periodic Riemann-Hilbert (MPR) problem that characterizes the OTPs. Through a sequence of transformations tailored to the cylindrical geometry and a subsequent nonlinear steepest descent analysis, we derive explicit asymptotic formulae. This work not only yields the desired asymptotics but also establishes a general Riemann-Hilbert framework for problems on periodic domains.
