Application of complex analysis to second order equations of mixed type

• Autorzy: Guo Chun Wen
• Języki publikacji: EN

Abstrakty

EN

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 $u_{xx} + sgn y u_{yy} = 0$ was investigated. In [3], the Tricomi problem for the generalized Lavrent'ev-Bitsadze equation $u_{xx} + sgn y u_{yy} + Au_x + Bu_y + Cu = 0$, i.e. $u_{ξη} + au_ξ + bu_η + cu = 0$ with the conditions: a ≥ 0, $a_ξ + ab - c ≥ 0$, 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.

Słowa kluczowe

EN

discontinuous Poincaré problem; equations of mixed type; complex analytic method

Kategorie tematyczne

• 35M10: Equations of mixed type

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1998
• Tom: 70
• Numer: 1
• Strony: 221-231

Daty

wydano

1998

otrzymano

1997-12-28

poprawiono

1998-08-31

Twórcy

autor

Guo Chun Wen

• Department of Mathematics, Peking University, Beijing 100871, China

Bibliografia

• [1] A. V. Bitsadze, Differential Equations of Mixed Type, MacMillan, New York, 1964.
• [2] A. V. Bitsadze, Some Classes of Partial Differential Equations, Gordon and Breach, New York, 1988.
• [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).
• [4] G. C. Wen, Conformal Mappings and Boundary Value Problems, Amer. Math. Soc., Providence, R.I., 1992, 137-188.
• [5] G. C. Wen, Oblique derivative problems for linear mixed equations of second order, Sci. in China Ser. A 41 (1998), 346-356.
• [6] G. C. Wen and H. Begehr, Boundary Value Problems for Elliptic Equations and Systems, Longman, Harlow, 1990, 217-272.