Oblique derivative problem for general Chaplygin-Rassias equations |
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| Authors: | Wen GuoChun |
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| Institution: | (1) LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, China |
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| Abstract: | The present paper deals with the oblique derivative problem for general second order equations of mixed (elliptic-hyperbolic)
type with the nonsmooth parabolic degenerate line in any plane domain D with the boundary ∂D=Γ ∪ L
1 ∪ L
2 ∪ L
3 ∪ L
4, where Γ(⊂ {y > 0}) ∈ C
μ
2
(0 < μ < 1) is a curve with the end points z = −1, 1. L
1, L
2, L
3, L
4 are four characteristics with the slopes −H
2(x)/H
1(y), H
2(x)/H
1(y),−H
2(x)/H
1(y),H
2(x)/H
1(y) (H
1(y) = √|K
1(y)|, H
2(x) = √|K
2(x)| in {y < 0}) passing through the points z = x + iy = −1, 0, 0, 1 respectively. And the boundary condition possesses the form in which z
1, z
2 are the intersection points of L
1, L
2, L
3, L
4 respectively. The above equations can be called the general Chaplygin-Rassias equations, which include the Chaplygin-Rassias
equations as their special case. The above boundary value problem includes the Tricomi problem of the Chaplygin equation: K(y)u
xx+u
yy = 0 with the boundary condition u(z) = ϕ(z) on Γ ∪ L
1 ∪ L
4 as a special case. Firstly some estimates and the existence of solutions of the corresponding boundary value problems for
the degenerate elliptic and hyperbolic equations of second order are discussed. Secondly, the solvability of the Tricomi problem,
the oblique derivative problem and Frankl problem for the general Chaplygin-Rassias equations are proved. The used method
in this paper is different from those in other papers, because the new notations W(z) = W(x + iy) =
= H
1(y)u
x − iH
2(x)u
y]/2 in the elliptic domain and W(z) = W(x+jy) =
=H
1(y)u
x − jH
2(x)u
y]/2 in the hyperbolic domain are introduced for the first time, such that the second order equations of mixed type can be
reduced to the mixed complex equations of first order with singular coefficients. And thirdly, the advantage of complex analytic
method is used, otherwise the complex analytic method cannot be applied.
This work was supported by the National Natural Science Foundation of China (Grant No. 10671207) |
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| Keywords: | oblique derivative problem equations of mixed type nonsmooth degenerate line |
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![$$\frac{1}{2}\frac{{\partial u}}{{\partial \nu }} = \frac{1}{{H(x,y)}}\operatorname{Re} \left {\overline {\lambda (z)} u_{\tilde z} } \right] = r(z), z \in \Gamma \cup L_1 \cup L_4 , \operatorname{Im} \left {\overline {\lambda (z)} u_{\tilde z} } \right]\left| {_{z = z_l } } \right. = b_l ,l = 1,2, u( - 1) = b_0 ,u(1) = b_3 ,$$](/content/b6123u118272987k/11425_2007_Article_153_TeX2GIFE2.gif)
