Application of complex analysis to second order equations of mixed type

Guo 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 -