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The mixed boundary value problem for the inhomogeneous Cimmino system. (English) Zbl 1312.35040
Summary: In this article, we first propose a kind of mixed boundary value problem for the inhomogeneous Cimmino system, which consists of first order linear partial differential equations in $$\mathbb{R}^{4}$$. Then, by using the one-to-one correspondence between the theory of quaternion valued hyperholomorphic functions and that of Cimmino system’s solutions, we transform the problem as stated above into a problem related to the $$\psi$$-hyperholomorphic functions in quaternionic analysis. Moreover, we show the boundedness, Hölder continuity, and generalized derivatives of a kind of singular integral operator $$^{\psi }T_{\mathbb{C}^{2}}[g]$$ related to $$\psi$$-hyperholomorphic functions in quaternionic analysis. Lastly, the solution of the mixed boundary value problem for the inhomogeneous Cimmino system is explicitly described.

##### MSC:
 35F05 Linear first-order PDEs
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