Riemann-Hilbert problems for poly-Hardy space on the unit ball

In this paper, we focus on a Riemann–Hilbert boundary value problem (BVP) with a constant coefficients for the poly-Hardy space on the real unit ball in higher dimensions. We first discuss the boundary behaviour of functions in the poly-Hardy class. Then we construct the Schwarz kernel and the higher order Schwarz operator to study Riemann–Hilbert BVPs over the unit ball for the poly- Hardy class. Finally, we obtain explicit integral expressions for their solutions. As a special case, monogenic signals as elements in the Hardy space over the unit sphere will be reconstructed in the case of boundary data given in terms of functions having values in a Clifford subalgebra. Such monogenic signals represent the generalization of analytic signals as elements of the Hardy space over the unit circle of the complex plane

Boundary Value Problem for Matrix Analogue of Helmholtz’s Equation (Poincaré’s Problem)

En este artículo se estudia un sistema de ecuaciones en derivadas parciales elípticas en el espacio complejo, considerado como un análogo matricial de la ecuación de Helmholtz. Se investiga cómo reducir el problema de Poincaré asociado a este sistema a un sistema integral singular equivalente, bajo ciertas condiciones en la frontera y en el infinito