Application of complex analysis to second order equations of mixed type

Guo Chun Wen

Guo Chun Wen

Annales Polonici Mathematici, Tome 69 (1998), p. 221-231 / Harvested from The Polish Digital Mathematics Library

Résumé

This paper deals with an application of complex analysis to second order equations of mixed type. We mainly discuss the discontinuous Poincaré boundary value problem for a second order linear equation of mixed (elliptic-hyperbolic) type, i.e. the generalized Lavrent’ev-Bitsadze equation with weak conditions, using the methods of complex analysis. We first give a representation of solutions for the above boundary value problem, and then give solvability conditions via the Fredholm theorem for integral equations. In [1], [2], the Dirichlet problem (Tricomi problem) for the mixed equation of second order $u_{x x} + s g n y u_{y y} = 0$ was investigated. In [3], the Tricomi problem for the generalized Lavrent’ev-Bitsadze equation $u_{x x} + s g n y u_{y y} + A u_{x} + B u_{y} + C u = 0$ , i.e. $u_{ξ η} + a u_{ξ} + b u_{η} + c u = 0$ with the conditions: a ≥ 0, $a_{ξ} + a b - c \geq 0$ , c ≥ 0 was discussed in the hyperbolic domain. In the present paper, we remove the above assumption of [3] and obtain a solvability result for the discontinuous Poincaré problem, which includes the corresponding results in [1]-[3] as special cases.

Publié le : 1998-01-01

Zbl 0923.35104

EUDML-ID : urn:eudml:doc:262769

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     author = {Guo Chun Wen},
     title = {Application of complex analysis to second order equations of mixed type},
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     year = {1998},
     pages = {221-231},
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Guo Chun Wen. Application of complex analysis to second order equations of mixed type. Annales Polonici Mathematici, Tome 69 (1998) pp. 221-231. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv70z1p221bwm/

Bibliographie

[000] [1] A. V. Bitsadze, Differential Equations of Mixed Type, MacMillan, New York, 1964. | Zbl 0111.29205

[001] [2] A. V. Bitsadze, Some Classes of Partial Differential Equations, Gordon and Breach, New York, 1988. | Zbl 0749.35002

[002] [3] S. P. Pul'kin, The Tricomi problem for the generalized Lavrent'ev-Bitsadze equation, Dokl. Akad. Nauk SSSR 118 (1958), 38-41 (in Russian).

[003] [4] G. C. Wen, Conformal Mappings and Boundary Value Problems, Amer. Math. Soc., Providence, R.I., 1992, 137-188.

[004] [5] G. C. Wen, Oblique derivative problems for linear mixed equations of second order, Sci. in China Ser. A 41 (1998), 346-356. | Zbl 0929.35099

[005] [6] G. C. Wen and H. Begehr, Boundary Value Problems for Elliptic Equations and Systems, Longman, Harlow, 1990, 217-272.