Boundary value problems for a second‐order elliptic partial differential equation system in Euclidean space

Abstract

Let be a bounded regular domain, let be the standard Dirac operator in , and let be the Clifford algebra constructed over the quadratic space . For fixed, denotes the space of ‐vectors in . In the framework of Clifford analysis, we consider two boundary value problems for a second‐order elliptic system of partial differential equations of the form in , where is a smooth ‐vector valued function. The boundary conditions of the problems contain the inner and outer products of the ‐vector solution with both the Dirac operator and the normal vector to , ensuring the well‐posedness for the problems. Investigation of the spectral properties of the sandwich operator is considered by using the Fredholm theory. Finally, it is shown that satisfactory problem‐solving properties, in general, fail when we replace the standard Dirac operator by those, obtained via unusual orthogonal bases of .

∂¯‐problem for a second‐order elliptic system in Clifford analysis

“…The flexibility introduced by structural sets ensembles makes it possible to seek new perspectives in investigations concerning the mapping properties of multidimensional Ahlfors–Beurling transforms and additive decompositions of harmonic functions [16, 19–26]. This permits us to consider a much wider class of second‐order partial differential equation from two given structural sets $$$ \nu $$$ and $$$ \vartheta $$$, that is, the general sandwich equation $$$$ {}^{\nu }{\partial}_{\underset{\_}{x}}{f}^{\vartheta }{\partial}_{\underset{\_}{x}}=0, $$$$ whose solutions have been referred to as $$$ \left(\nu, \vartheta \right) $$$‐inframonogenic functions (see prior research [5, 11, 27–30]).…”