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| THE COMPLEX FORM AND SOME BOUNDARYVALUE PROBLEMS FOR NONLINEAR ELLIPTICSYSTEMS OF SECOND ORDER |
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Citation： |
WEN GIJOCHUN,FANG AINONG.THE COMPLEX FORM AND SOME BOUNDARYVALUE PROBLEMS FOR NONLINEAR ELLIPTICSYSTEMS OF SECOND ORDER[J].Chinese Annals of Mathematics B,1981,2(2):200~216 |
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Net amount： 818 |
Authors： |
WEN GIJOCHUN; FANG AINONG |
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| Abstract： |
The present paper is concerned with the nonlinear elliptic system of second order.
Firstly, we shall establish a complex form of the system. Secondly .we shall consider the solvability of some boundary value problems for tbe complex equation of second order.
let
(1) \[{\Phi _j}(x,y,U,V,{U_x},{U_y},...,{U_{xx}},{U_{yy}},{V_{xx}},{V_{xy}},{V_{yy}}) = 0,j = 1,2\] be the I. G. Petrowkii’s nonlinear elliptic system of second Qrder in the botinded domain G, where \[{\Phi _j}(x,y,{z_1},...,{z_{12}})(j = 1,2)\]) are continuous real functions of
the variables \[x,y[(x,y) \in G],{z_1},...,{z_{12}} \in R\], (the real axis), and contiriupusly
differentiable for \[{z_1},...,{z_{12}} \in R\]. The solutions \[[U(x,y),V(x,y)]\], F(a?, y)] of the system are understood in the generalized sense.
THEOBEM I. i) If the I. G. Petrovskii;s nonlinear system of equations (1) satisfies the M. I. visik-D. Xiagi’s uniformly elliptic condition for any solutions
U(x,y),V(x,y) of (1) in G, then it can be written as the following complex equation?
(2)\[{W_{z\overline z }} = F(z,W,{W_z},\overline {{W_z}} ,{W_{zz}},{\overline W _{zz}})\]
where W=U+iV, z=x+iy, \[{W_z} = \frac{1}{2}[{W_x} - i{W_y}],...,\],
ii) If the I. G. Petrovskii's nonlinear elliptic system (1) satisfies the condition that there exist two positive constants \[\delta \] and K, such that
(3) \[|{\Phi _{j{U_{xx}}}}|,|{\Phi _{j{U_{xy}}}}|,|{\Phi _{j{U_{yy}}}}|,|{\Phi _{j{V_{xx}}}}|,|{\Phi _{j{V_{xy}}}}|,|{\Phi _{j{V_{yy}}}}| \leqslant K,j = 1,2\]
\[|det(A)| \geqslant \delta > 0\], in G, then by a suitable linear trans-formation of the variables (x,y)into variables \[(\xi ,\eta )\], system (1) can be written as the following coinplex equation
⑷ \[{W_{\xi \xi }} = F(\xi ,W,{W_\xi },{\overline W _\xi },{W_{\xi \xi }},{\overline W _{\xi \xi }}),\varsigma = \xi + i\eta \]
In the following section, we discuss the complex equation (2) of the following
form:
,We^B(z9 Wee)x .\[(5)\left\{ \begin{gathered}
{W_{zz}} = F(z,W,{W_z},{\overline W _z},{W_{zz}},{\overline W _{zz}}) \hfill \ F = {Q_1}{W_{zz}} + {Q_2}\overline {{W_{\overline z \overline z }}} + {Q_4}{W_{zz}} + {A_1}{W_z} + {A_2}{\overline W _{\overline z }} \hfill \ + {A_3}\overline {{W_z}} + {A_4}{W_{\bar z}} + {A_5}W + {A_6}\bar W + {A_7}, \hfill \ {Q_j} = {Q_j}(z,W,{W_{\bar z}},{\overline W _{\bar z}},{W_{zz}},{\overline W _{zz}}),j = 1,...,4 \hfill \ {A_j} = {A_j}(z,W,{W_z},{\overline W _z}),j = 1,...,7 \hfill \\
\end{gathered} \right.\]
1) \[{Q_j}(z,W,{W_z},{\overline W _z},U,V),j = 1,...,4.{A_j} = (z,W,{W_z},{\overline W _z}),j = 1,...,7\] are measurable functions of z for any continuously differentiable functions W(z) and measurable functions U(z), V(z) in G, Furthermore they satisfy
(6)\[{\left\| {{A_j}} \right\|_{{L_p}(\overline {G)} }} \leqslant {K_0},j = 1,2,{\left\| {{A_j}} \right\|_{{L_p}(\overline {G)} }} \leqslant {K_1},j = 3,...,7\]
where\[{K_0},{K_1}( \leqslant {K_0}),p( > 2)\] are constants:
2) Qj, Aj are continuous for \[W,{W_z},{\overline W _z} \in E\](the whole plane) and the continuity is uniform with respect to almost every point \[z \in G\] and \[U,V \in E\]
3) \[F(z,W,{W_z},{\overline W _z},U,V)\] satisfies the following Lipschitz's condition, i.e. for
almost every point \[z \in G\], and for all \[W,{W_z},{\overline W _z}{U_1},{U_2},{V_1},{V_2} \in E\], the inequality
(7)\[\begin{gathered}
|F(z,W,{W_z},{\overline W _z},{U_1},{V_1}) - F(z,W,{W_z},{\overline W _z},{U_2},{V_2})| \hfill \ \leqslant {q_0}|{U_1} - {U_2}| + q_0^'|{V_1} - {V_2}|,{q_0} + q_0^' < 1 \hfill \\
\end{gathered} \] holds where \[{q_0},q_0^'\] are two nonnegative constants.
In this paper, let G be a simply connected domain with boundary \[\Gamma \in C_\mu ^2(0 < \mu < 1)\]； without loss of geaerality, we may assume that G is the unit disk |z|<1. Now
we, describe the results of the solvability of Riemann-Hilbert botindary value problem (Problem R-H) and the oblique derivative problem (Problem P) for Eq. (5) in the unit disk G: |z| <1.
Problem R-H. We try to find a solution W（z）of Eq. (5) which is continuonsly differentiable on \[G\], and satisfies the boundary conditions:
(8) \[\operatorname{Re} [{{\bar z}^{{\chi _1}}},{W_z}] = {r_1}(z),Re[{{\bar z}^{{\chi _2}}}\overline {W(z)} ] = {r_2}(z),z \in \Gamma \]?
where \[{\chi _1},{\chi _2}\] are two integers, and \[{r_j} \in C_v^{j - 1}(\Gamma ),j = 1,2,\frac{1}{2} < v < 1\]
Problem P. we try to find a solution W（z) of Eq. (5) which is continuously
diffierentiabfe on \[\overline G \] and satisfies the boundaory conditions：
(9) \[\operatorname{Re} [{{\bar z}^{{\chi _1}}}{W_z}] = {r_1}(z),Re[{{\bar z}^{{\chi _2}}}\overline {W(z)} ] = {r_2}(z),z \in \Gamma \],
Where \[{\chi _1},{\chi _2},{r_1}(z),{r_2}(z)\] are the same as in (8), but \[{r_2}(z) \in {C_v}(\Gamma )\].
Theorem II. Suppose that Eq. (5) satisfies the condition C and the constants \[q_0^'\] and K1 are adequately small; then the solvability of Problem R-H is as follows：
1) When \[{\chi _1} \geqslant 0,{\chi _2} \geqslant 0\] Problem R-H is solvable;
2) When \[{\chi _1} < 0,{\chi _2} \geqslant 0(or{\kern 1pt} {\kern 1pt} {\chi _1} \geqslant 0,{\chi _2} < 0){\kern 1pt} \] there are \[2|{\chi _1}| - 1(or2|{\chi _2}| - 1)\] solvable
conditions for Problem R-H;
3) WHen \[{\chi _1} < 0,{\chi _2} < 0\], there are \[2(|{\chi _1}| + |{\chi _2}| - 1)\] solvable conditions for Problem
R-H.
Theorem III Let Eq (5) satisf the condition C and the constants \[q_0^'\] and \[{K_1}\] are adequately small, then tbe solvability of Problem P is as follows：
1) When \[{\chi _1} \geqslant 0,{\chi _2} \geqslant 0\] Problem P is solvable;
2) When \[{\chi _1} < 0,{\chi _2} \geqslant 0(or{\kern 1pt} {\kern 1pt} {\chi _1} \geqslant 0,{\chi _2} < 0){\kern 1pt} {\kern 1pt} {\kern 1pt} \], there are \[2|{\chi _1}| - 1(or2|{\chi _2}| - 1)\] solvable conditions for Problem P;
3) When \[{\chi _1} < 0,{\chi _2} < 0\]; there are \[2|{\chi _1}|{\text{ + }}|{\chi _2}| - 1)\] solvable conditions for Problem P.
Furthermore, the solution W(z) of Problem P for Eq. (5) may be expressed as
\[{g_j}(\xi ,z) = \left\{ \begin{gathered}
\int_0^z {\frac{{{z^{2{\chi _j} + 1}}}}{{1 - \bar \xi z}}dz,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} for{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\chi _j} \geqslant 0} \hfill \ \int_0^z {\frac{{{\xi ^{ - 2{\chi _j} - 1}}}}{{1 - \bar \xi z}}dz,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} for{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\chi _j} < 0} \hfill \\
\end{gathered} \right.j = 1,2\]
where \[{\Phi _0}(z) = a + ib\] is a complex constant,and \[{\Phi _1}(z),{\Phi _2}(z)\] are two analytic functions.
The proofs of the above stated theorems are based on a prior estimates for the bounded solutes of these boundary value problems and Leray-Schander theorem.
Besides, we have considered also the solvability of Problem R-H and Problem P for Eq. (6) in the multiply connected domain. |
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