论文标题

半线性beltrami方程理论

To the theory of semi-linear Beltrami equations

论文作者

Gutlyanskii, V., Nesmelova, O., Ryazanov, V., Yakubov, E.

论文摘要

本文致力于研究半线性Beltrami方程,这些方程与相应的半线性泊松泊松型在平面上数学物理学的半线性泊松型方程在各向异性和不均匀培养基中。 在其第一部分中,通过Ahlfors-bers和Leray-schauder方法应用完全连续的OPE \ -RA \ -tors,我们证明存在没有边界条件的半线性Beltrami方程的​​常规解。此外,在这里,我们通过VEKUA类型方程的解决方案和具有源的广义分析函数得出它们的表示。 结果,将这些结果的一系列应用在半线性泊松类型方程中以及数学物理学的相应方程式描述,这些方程描述了与物理和化学吸收,血浆状态以及在各向异性和非正及媒介中的物理和化学吸收,血浆状态和固定燃烧的现象。 本文的第二部分包含了半线性Beltrami方程的​​希尔伯特(Dirichlet)边界价值问题的存在,表示和规律性结果,以及针对与众人相关能力的任意边界数据的半线性Poisson类型方程的PoinCare(Neumann)边界价值问题。

The present paper is devoted to the study of semi-linear Beltrami equations which are closely relevant to the corresponding semi-linear Poisson type equations of mathematical physics on the plane in anisotropic and inhomogeneous media. In its first part, applying completely continuous ope\-ra\-tors by Ahlfors-Bers and Leray--Schauder approach, we prove existence of regular solutions of the semi-linear Beltrami equations with no boundary conditions. Moreover, here we derive their representation through solutions of the Vekua type equations and generalized analytic functions with sources. As consequences, it is given a series of applications of these results to semi-linear Poisson type equations and to the corresponding equations of mathematical physics describing such phenomena as diffusion with physical and chemical absorption, plasma states and stationary burning in anisotropic and inhomogeneous media. The second part of the paper contains existence, representation and regularity results for nonclassical solutions to the Hilbert (Dirichlet) boundary value problem for semi-linear Beltrami equations and to the Poincare (Neumann) boundary value problem for semi-linear Poisson type equations with arbitrary boundary data that are measurable with respect to logarithmic capacity.

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