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1998 | 70 | 1 | 221-231

Tytuł artykułu

Application of complex analysis to second order equations of mixed type

Autorzy

Treść / Zawartość

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.

Kategorie tematyczne

Rocznik

Tom

70

Numer

1

Strony

221-231

Daty

wydano
1998
otrzymano
1997-12-28
poprawiono
1998-08-31

Twórcy

autor
  • 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.

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