By Guo Chun Wen

ISBN-10: 9812779426

ISBN-13: 9789812779427

Within the fresh half-century, many mathematicians have investigated a variety of difficulties on a number of equations of combined kind and bought attention-grabbing effects, with very important functions to fuel dynamics. besides the fact that, the Tricomi challenge of basic combined style equations of moment order with parabolic degeneracy has now not been thoroughly solved, quite the Tricomi and Frankl difficulties for basic Chaplygin equation in multiply attached domain names posed by means of L Bers, and the life, regularity of strategies of the above difficulties for combined equations with non-smooth degenerate curve in numerous domain names posed through J M Rassias.

the strategy published during this publication is in contrast to the other, within which the hyperbolic quantity and hyperbolic advanced functionality in hyperbolic domain names, and the complicated quantity and intricate functionality in elliptic domain names are used. The corresponding difficulties for first order advanced equations with singular coefficients are first mentioned, after which the issues for moment order advanced equations are thought of, the place we pose the hot partial by-product notations and complicated analytic tools such that the different types of the above first order complicated equations in hyperbolic and elliptic domain names are fully exact. meanwhile, the estimates of ideas for the above difficulties are bought, therefore many open difficulties together with the above Tricomi Bers and Tricomi Frankl Rassias difficulties will be solved.

Contents: Elliptic advanced Equations of First Order; Elliptic advanced Equations of moment Order; Hyperbolic complicated Equations of First and moment Orders; First Order advanced Equations of combined style; moment Order Linear Equations of combined style; moment Order Quasilinear Equations of combined kind.

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6) Elliptic, Hyperbolic and Mixed Complex Equations 44 where C1 (ζ) = −2(ln Ψ)ζ¯ + B1 . e. e. 1 ¯ (Vξξ + Vηη ) = 0 in G, 4 in which ζ = ξ + iη (see [86]9)). 7) where Z = {−1, 1} is the set of discontinuous points of λ(z) on Γ∗ , ν is a given vector at every point on Γ∗ , λ(z) = a(x) + ib(x) = cos(ν, x) − i cos(ν, y), cos(ν, n) ≥ 0 on Γ∗ , here n is the outward normal vector at every point on Γ∗ , b0 , b1 are real constants, and λ(z), r(z), b0 , b1 satisfy the conditions Cα [λ(z), Γ∗ ] ≤ k0 , Cα [r(z), Γ∗ ] ≤ k2 , |b0 |, |b1 | ≤ k2 .

5) satisfies Condition C. 10), and t1 = −1, t2 = 1, δ is a sufficiently Chapter I Elliptic Complex Equations of First Order 19 small positive constant, k = (k0 , k1 , k2 ), and M2 = M2 (δ, k, H, DZ ) is a non-negative constant.. 13). 21) ∗ in which z(Z) is the inverse function of Z(z), Z0 = Z(z0 ), ∂DZ = ∂DZ \ ˜ on ∂DZ is K = 0. 20). This completes the proof. 5) satisfies Condition C. 13), W (Z) = w[z(Z)] − ψ(Z), Z = x + iY = x + iG(y), and Φ[z(Z)] is an analytic function in DZ . 5) in the form ψ(Z) = − 1 π D0 f (t) dσt , H(y)f (Z) ∈ L∞ (DZ ).

E. η = 1, then we can choose β = m/(m + 2) − δ, δ is a sufficiently small positive constant. 5). 5) satisfies Condition C. 10), and t1 = −1, t2 = 1, δ is a sufficiently Chapter I Elliptic Complex Equations of First Order 19 small positive constant, k = (k0 , k1 , k2 ), and M2 = M2 (δ, k, H, DZ ) is a non-negative constant.. 13). 21) ∗ in which z(Z) is the inverse function of Z(z), Z0 = Z(z0 ), ∂DZ = ∂DZ \ ˜ on ∂DZ is K = 0. 20). This completes the proof. 5) satisfies Condition C. 13), W (Z) = w[z(Z)] − ψ(Z), Z = x + iY = x + iG(y), and Φ[z(Z)] is an analytic function in DZ .

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Elliptic, Hyperbolic and Mixed Complex Equations with Parabolic Degeneracy: Including Tricomi-Bers and Tricomi-Frankl-Rrassias Problems (Peking University Series in Mathematics) by Guo Chun Wen


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