OBLIQUE DERIVATIVE PROBLEMS FOR SECOND-ORDER HYPERBOLIC EQUATIONS WITH DEGENERATE CURVE

Guo-Chun Wen
2011 Electronic Journal of Differential Equations   unpublished
The present article concerns the oblique derivative problem for second order hyperbolic equations with degenerate circle arc. Firstly the formulation of the oblique derivative problem for the equations is given, next the representation and estimates of solutions for the above problem are obtained, moreover the existence of solutions for the problem is proved by the successive iteration of solutions of the equations. In this article, we use the complex analytic method, namely the new partial
more » ... the new partial derivative notations, hyperbolic complex functions are introduced, such that the second order hyperbolic equations with degenerate curve are reduced to the first order hyperbolic complex equations with singular coefficients, then the advantage of complex analytic method can be applied. 1. Formulation of the oblique derivative problem In [1, 2, 3, 4, 5, 8, 9, 10], the authors posed and discussed the Cauchy problem, Dirichlet problem and oblique derivative boundary value problem of second order hyperbolic equations and mixed equations with parabolic degenerate straight lines by using the methods of integral equations, functional analysis, energy integrals, complex analysis and so on, the obtained results possess the important applications. Here we generalize the above results to the oblique derivative problem of hyperbolic equations with degenerate circle arc. In this article, the used notations are the same as in [6, 7, 8, 9, 10]. Let D be a simply connected bounded domain D in the hyperbolic complex plane C with the boundary ∂D = L ∪ L 0 , where L = L 1 ∪ L 2. Herein and later on, denotêdenotê y = y − √ R 2 − x 2 , and L 1 = {x + G(ˆ y) = R * , x ∈ [R * , 0]}, L 2 = {x − G(ˆ y) = R * , x ∈ [0, R * ]}, L 0 = {R * ≤ x ≤ R * , ˆ y = 0}, in which K(ˆ y) = −|ˆy||ˆy| m , m, R are positive numbers, R * = −R, R * = R, z 0 = z 1 = jy 0 = jy 1 the intersection of L 1 , L 2 , G(ˆ y) = ˆ y 0 |K(t)|dt, H(ˆ y) = |K(ˆ y)| 1/2. In this article we use the hyperbolic unit j with the condition j 2 = 1 in D, and x + jy, w(z) = U (z) + jV (z) = [H(ˆ y)u x − ju y ]/2 are called the hyperbolic number 2000 Mathematics Subject Classification. 35L20, 35L80.
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