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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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