# Application of complex analysis to second order equations of mixed type

Annales Polonici Mathematici (1998)

- Volume: 70, Issue: 1, page 221-231
- ISSN: 0066-2216

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topGuo Chun Wen. "Application of complex analysis to second order equations of mixed type." Annales Polonici Mathematici 70.1 (1998): 221-231. <http://eudml.org/doc/262769>.

@article{GuoChunWen1998,

abstract = {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.},

author = {Guo Chun Wen},

journal = {Annales Polonici Mathematici},

keywords = {discontinuous Poincaré problem; equations of mixed type; complex analytic method; discontinuous Poincaré boundary value problem; second-order linear equation of mixed (elliptic-hyperbolic) type; generalized Lavrent'ev-Bitsadze equation; Fredholm theorem for integral equations},

language = {eng},

number = {1},

pages = {221-231},

title = {Application of complex analysis to second order equations of mixed type},

url = {http://eudml.org/doc/262769},

volume = {70},

year = {1998},

}

TY - JOUR

AU - Guo Chun Wen

TI - Application of complex analysis to second order equations of mixed type

JO - Annales Polonici Mathematici

PY - 1998

VL - 70

IS - 1

SP - 221

EP - 231

AB - 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.

LA - eng

KW - discontinuous Poincaré problem; equations of mixed type; complex analytic method; discontinuous Poincaré boundary value problem; second-order linear equation of mixed (elliptic-hyperbolic) type; generalized Lavrent'ev-Bitsadze equation; Fredholm theorem for integral equations

UR - http://eudml.org/doc/262769

ER -

## References

top- [1] A. V. Bitsadze, Differential Equations of Mixed Type, MacMillan, New York, 1964. Zbl0111.29205
- [2] A. V. Bitsadze, Some Classes of Partial Differential Equations, Gordon and Breach, New York, 1988. Zbl0749.35002
- [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. Zbl0929.35099
- [6] G. C. Wen and H. Begehr, Boundary Value Problems for Elliptic Equations and Systems, Longman, Harlow, 1990, 217-272.

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