ON LINEAR AND NONLINEAR RIEMANN-HILBERT PROBLEMS FOR REGULAR FUNCTION WITH VALUES IN A CLIFFORD ALGEBRA

This paper deals with the boundary value problems for regular function with valuesin a Clifford algebra: ()W=O, x∈R~n\Г, w~+(x)=G(x)W~-(x)+λf(x, W~+(x), W~-(x)), x∈Г; W~-(∞)=0,where Г is a Liapunov surface in R~n the differential operator ()=()/()x_1+()/()x_2+…+()/()x_ne_n, W(x) =∑_A, ()_AW_A(x) are unknown functions with values in a Clifford algebra ()_n Undersome hypotheses, it is proved that the linear baundary value problem (where λf(x, W~+(x),W~-(x)) =g(x)) has a unique solution and the nonlinear boundary value problem has atleast one solution.

On Linear and Nonlinear Riemann-Hilbert Problems For Regular Function with Values in a Clifford Algebra

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.