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| On Linear and Nonlinear Riemann-Hilbert Problems For Regular Functionwith Values in a Clifford Algebra |
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Citation： |
Xu Zhenyuan.On Linear and Nonlinear Riemann-Hilbert Problems For Regular Functionwith Values in a Clifford Algebra[J].Chinese Annals of Mathematics B,1990,11(3):349~358 |
| Page view： 752
Net amount： 843 |
Authors： |
Xu Zhenyuan; |
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| Abstract： |
This paper deals with the boundary value problems for regular function with values, in a Clifford algebra:
$[\begin{array}{l}
\bar \partial W = 0,x \in {R^n}\backslash \Gamma ,\{W^ + }(x) = G(x){W^ - }(x) + \lambda f(x,{W^ + }(x),{W^ - }(x)),x \in \Gamma ;{W^ - }(\infty ) = 0
\end{array}\]$
where \Gamma is a Liapunov1 surface in R^n, the differential operator $[\bar \partial = \frac{\partial }{{\partial {x_1}}} + \frac{\partial }{{\partial {x_2}}}{e_2} + \cdots + \frac{\partial }{{\partial {x_n}}}{e_n},W(x) = \sum\limits_\Lambda {{e_A}{W_A}(x)} \]$ are unknown functions with values in a Clifford algebra \mathscr{A}_n. Under some hypotheses, it is proved that the. linear baundary value problem (where \lambda f(x,W^+(x), W^-(x))\equiv g(x)) has a unique solution and the nonlinear boundary value problem has at least one solution. |
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