The Tricomi and Frankl problems for generalized Chaplygin equations in multiply connected domains. (English) Zbl 1173.35612
The generalized Chaplygin equation
\[ L_u=K(y)u_{xx}+u_{yy}+ a(x,y)u_x+b(x,y)u_y+c(x,y)u=-d(x,y) \]
is considered in multiply connected domain \(D\). It is supposed that the coefficients of the equation satisfy
\[ L_\infty[\eta,\overline D^+]\leq k_0, \eta =a,b,c,L_\infty[d,\overline D^+]\leq k_1,c\leq 0\text{ in }\overline D^+ \]
\[ \widetilde C[d,\overline D^-]=C[d,\overline D^-]+C[d_x,\overline D^-] \leq k_1, \widetilde C[\eta,\overline D^-]\leq k_0,\eta=a,b,c, \]
where \(k_0\), \(k_1\) are positive constants. In a case \(a=b=c=d=0\) the above equation is the so-called Chaplygin equation.
The Tricomi and Frankl problems for generalized Chaplygin equation in multiply connected domain are under consideration in the paper. The representation of solutions of the Tricomi problem for the equation is given, and the uniqueness and existence of solutions for the problem by a new method are proved using the complex functions in the elliptic domain and the hyperbolic complex functions in hyperbolic domain.
\[ L_u=K(y)u_{xx}+u_{yy}+ a(x,y)u_x+b(x,y)u_y+c(x,y)u=-d(x,y) \]
is considered in multiply connected domain \(D\). It is supposed that the coefficients of the equation satisfy
\[ L_\infty[\eta,\overline D^+]\leq k_0, \eta =a,b,c,L_\infty[d,\overline D^+]\leq k_1,c\leq 0\text{ in }\overline D^+ \]
\[ \widetilde C[d,\overline D^-]=C[d,\overline D^-]+C[d_x,\overline D^-] \leq k_1, \widetilde C[\eta,\overline D^-]\leq k_0,\eta=a,b,c, \]
where \(k_0\), \(k_1\) are positive constants. In a case \(a=b=c=d=0\) the above equation is the so-called Chaplygin equation.
The Tricomi and Frankl problems for generalized Chaplygin equation in multiply connected domain are under consideration in the paper. The representation of solutions of the Tricomi problem for the equation is given, and the uniqueness and existence of solutions for the problem by a new method are proved using the complex functions in the elliptic domain and the hyperbolic complex functions in hyperbolic domain.
Reviewer: Elena Gavrilova (Sofia)
MSC:
| 35M10 | PDEs of mixed type |
| 35J70 | Degenerate elliptic equations |
| 35L80 | Degenerate hyperbolic equations |
| 35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |
| 35C05 | Solutions to PDEs in closed form |
Keywords:
tricomi and frankl problems; generalized chaplygin equations; multiply connected domains; complex functions; hyperbolic complex functionsReferences:
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