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Existence of solutions to a multidimensional analog of the Beltrami equation. (English. Russian original) Zbl 0695.35137
Sib. Math. J. 30, No. 1, 79-87 (1989); translation from Sib. Mat. Zh. 30, No. 1(173), 103-113 (1989).
Let $$U\subset C^ n$$, $$\mu$$ : $$U\to$$ a set of complex $$n\times n$$ matrices with measurable elements $$\mu_{kj}(z)$$ $$(k,j=1,...,n)$$, belonging to the space $$L^{\infty}(U)$$. In the paper for the following multidimensional analog of Beltrami equation: $(*)\quad {\bar \partial}_ kf(z)=\sum^{n}_{j=1}\partial_ jf(z)\mu_{jk}(z)\quad (k=1,...,n),$ where f: $$U\to C$$ is a solution of (*) with general (in the Sobolev sense) derivatives $${\bar \partial}_ kf$$, $$\partial_ kf$$ $$(k=1,...,n)$$, a theorem on existence and uniqueness of the solution of the equation (*) is proved.
Reviewer: W.Kotarski
MSC:
 35N10 Overdetermined systems of PDEs with variable coefficients 35F05 Linear first-order PDEs
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References:
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