| Abstract: | In this paper, we consider the generalized Riemann-Hilberij problem for second
order quasi-linear elliptic complex equation
\\begin{array}{l}
\frac{{{\partial ^2}w}}{{\partial {{\bar z}^2}}} + {q_1}(z,w,\frac{{\partial w}}{{\partial \bar z}},\frac{{\partial w}}{{\partial z}})\frac{{{\partial ^2}w}}{{\partial {z^2}}} + {q_2}(z,w,\frac{{\partial w}}{{\partial \bar z}},\frac{{\partial w}}{{\partial z}})\frac{{{\partial ^2}\bar w}}{{\partial z\partial \bar z}}\{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + {q_3}(z,w,\frac{{\partial w}}{{\partial \bar z}},\frac{{\partial w}}{{\partial z}})\frac{{{\partial ^2}w}}{{\partial z\partial \bar z}} + {q_4}(z,w,\frac{{\partial w}}{{\partial \bar z}},\frac{{\partial w}}{{\partial z}})\frac{{{\partial ^2}\bar w}}{{\partial z\partial \bar z}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (1)\{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \gamma (z,w,\frac{{\partial w}}{{\partial \bar z}},\frac{{\partial w}}{{\partial z}}),z \in G
\end{array}\]
satifying the boundary condition
\{\mathop{\rm Re}\nolimits} \left {{{\bar \lambda }_1}(z)\frac{{\partial w}}{{\partial \bar z}}} \right] = {\gamma _1}(z),{\mathop{\rm Re}\nolimits} \left {{{\bar \lambda }_2}(z)\frac{{\partial w}}{{\partial \bar z}}} \right] = {\gamma _2}(z),z \in \gamma {\kern 1pt} {\kern 1pt} {\kern 1pt} (2)\]
Many authors (see that papers 1, 4-6) have studied the Diriohlet problem and Riemann-Hilbert problem for linear elliptic complex equation. In our papers 2, 3 we also
considered the generalized Riemann-Hilbert problem of the general second order linear elliptic complex equation. We obtained the existence theorem, the explicit form of
generalized solution and the sufficient and necessary conditions for the solvability of the above mentioned boundary value problem.
Based on these results and applying the property of the introduced integral operators
and Schauder's fixed-point principle, it can be proved that the analogous deductions in 3 also hold for the generalized Riemann-Hilber problem (1), (2) of the quasi-linear complex equation, i, e., we have the following theorem:
Theorem, If the coefficients of second order quasi-linear elliptic complex equation
(1) satifies some conditions then
i) When index \({n_1} \ge 0,{n_2} \ge 0\), the boundary value problem (1), (2)
is always solvable and the solution depends on 2 \(2({n_1} + {n_2} + 1)\) arbitrary real constants.
ii) When index \({n_1} \ge 0,{n_2} < 0{\kern 1pt} {\kern 1pt} {\kern 1pt} (or{\kern 1pt} {\kern 1pt} {\kern 1pt} {n_1} < 0,{n_2} \ge 0{\kern 1pt} )\), the sufficient and necessary condition for the solvability of the above mentioned boundary value problem (1),(2) consists of \( - 2{n_2} - 1{\kern 1pt} {\kern 1pt} {\kern 1pt} ( - 2n, - 1)\) real equalities, if and only if the equalities are
satisfied, the boundary value problem is solvable and the solution depends on
\(2{n_1} + 1{\kern 1pt} {\kern 1pt} (2{n_2} + 1)\) arbitrary real constants.
iii)When index \({n_1} < 0,{n_2} < 0\), the sufficient and necessary condition for the solvability of the above mentioned boundary value problem (1) , (2) consists of
\( - 2({n_1} + {n_2} + 1)\) real equalities, if and only if the equalitieis are satisfied, the boundary-value problem is solvable.
Finally, in the similar way, we may farther extend the result to the case of the
nonlinear uniform elliptic complex equation. |
