Two Kinds of Boundary Value Problems for the First Order Systems of Nonlinear Elliptic Partial Differential Equations

In this paper we establish the esxistence and uniqueness theorems of a solution of the problem P(|z|=1,Rez^-nw_z]=0) and H(|z|=1,Rez^-nw]=0) for the nonlinear system (A):w_{\bar z}=g(z,w,w_z),|g(z,w,w_z^1)-g(z,w,w_z^2)| \leq q_0|w_z^1-w_z^2|,q_0=const.<1,z \in G={|z|<1}.for the problem P,where w(z) \in C_\alpha^1(\bar G),let C_\alpha(g(z,w,t)-g(z,w_1,t_1),\bar G) \leq H_3C_\alpha(w-w_1,\bar G)+H_4C_\alpha(t-t_1,\bar G),H_3,H_4 and C_\alpha(g(z,0,0),\bar G)be the suitable small constants.For the problem H,where w(z) \inW_p^(1)(\bar G),p>2,let |g(z,w,0)| \leq A(z,w)|w|^m+B(z,w),m>1,A(z,w) \in L_p(\bar G) and ||B(z,w)||_{L_p(\bar G)} be the suitable small.