Singular points of solutions of some elliptic systems on the plane

Makatsaria, G.

doi:10.1007/s10958-009-9525-9

Cited by 6 publications

(2 citation statements)

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“…Approximately in the same period Bers, independently from Vekua, suggested a generalization of analytic functions (so-called pseudo-analytic functions), based on the modification of the concept of the derivative. Note that many authors have proposed various generalizations, reducing system (4) to particular cases, until the complete theory of generalized analytic functions emerged[see [20][21][22][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]].…”

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confidence: 99%

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On the nonlinear Cauchy-Riemann equations of structural transformation and nonlinear Laplace equation

Wang

2018

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A function κ-transformation w (z) → w (z) = w (z) (1 + κ (z)) is constructed with the initial condition κ (z 0 ) = 0 at the point z 0 ∈ Ω and a more general K-transformation w (z) → w (z) = w (z) K (z) that is also made to rediscuss the complex differentiability for a given complex function w and subsequently we obtain a certain continuity and differentiability new criteria-nonlinear K (z)-structural Cauchy-Riemann equations to judge a complex function to be complex structural differentiable, namely, K-structural holomorphic condition in C or C n respectively. Then we found an unique Carleman-Bers-Vekua equations which is more simpler that all coefficients are dependent to the structural function κ (z) or K (z). The generalized K (z)-exterior differential operator and the generalized structural Wirtinger derivatives are simultaneously obtained as well. Nonlinear K (z)-structural Laplace equation is studied in the form of a secondorder partial differential equation.In this framework of K (z)-structural function, we surprisely discover that structural function K (z) is innately dependent to the complex domain or complex manifold themself, and the Cauchy-Riemann equation or nonlinear Cauchy-Riemann equation or Carleman-Bers-Vekua equations can be unified and expressed in a simple and compact differential form-K (z)-structural holomorphic differential equation, since K (z) can be chosen arbitrarily, thus it has greatly generalized the applied practicability what we study.

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