부산대학교 도서관

The author considers the following model contact problem: In the half-space $x_3>0$ find a harmonic function $u(x)$ with $u(x)=O(|x|^{-1})$ for $|x|\to\infty$, a closed domain $S\subseteq\Omega\subset\bold R^2=\{x\colon x_3=0\}$, and a vector $h=(h_1,h_2,h_3)$ such that the following conditions are satisfied: $2\pi\lambda u(x)=g(x)$, $u_{x_3}(x)\geq 0$ for $x\in S$, $2\pi\lambda u(x)>g(x)$ for $x\in\Omega\sbs S$, $u_{x_3}(x)=0$ for $x\in\bold R^2\sbs S$, $\int_Su_{x_3}(x)dS_x=P$, $\int_Sx_2u_{x_3}(x)dS_x=M_x$, $\int_Sx_1u_{x_3}(x)dS_x=M_y$, $g(x)=h_1+h_2x_2+H_3x_1-f(x_1,x_2)$, $f(x_1,x_2)\geq 0$, $\lambda={\rm const}>0$. Here $f(x)\in L_q(\Omega)$, $1< q<4$, $P_0=(P,M_x,M_y)\in\bold R^3$, $P>0$, are given. By means of the potential $u(x)=(2\pi)^{-1}\int_S|x-y|^{-1}p(y)dS_y$, $x\in S$, the problem of determining $(p,S,h)$ is reduced to the equations $(*)$ $\mu Jv^-+\lambda Kv^+=g$, $Iv^+=P$, $Ix_2v^+=M_x$, $Ix_1v^+=M_y$, where $v^+(x)=\sup\{v(x),0\}$, $v^-(x)=\inf\{v(x),0\}$, $p=v^+$, $S=\{x\colon v\geq 0\}$, $J$ is the duality mapping of $L_r$, $K$ a strongly positive, compact, and selfadjoint operator from $L_r$ in $L_q$ for $\frac430$, for $w_\epsilon=(v_\epsilon,h_\epsilon)$ in the space $F=L_r(\Omega)\times{\bf R}^3$, $\|w\|_F=\|v\|_{L^r}+\|h\|_{\bold R^3}$. The system $(**)$ has a unique solution $w_\epsilon\in F$ for every $\epsilon>0$. The system $(*)$ has a unique solution $w=(v,h)\in F$ if and only if $\|w_\epsilon\|_F\leq c$ for all $\epsilon\in(0,\epsilon_0]$. In this case $\|w_\epsilon-w\|_F\to 0$ holds for $\epsilon\to 0$.