Pseudomonotone operators and nonlinear elliptic boundary value problems
Nobuyuki Kenmochi · Journal of the Mathematical Society of Japan · 1975
KENMOCHI$\left\{\begin{array}{ll}u|_{\Gamma}=\psi & on \Gamma_{0},\\\sum_{k=1}^{N}(A_{k}(x, u & abla u)|_{\Gamma}) u_{k}=\psi^{*}+h(u|_{\Gamma}) on \Gamma\backslash \Gamma_{0},\end{array}\right.$where $\Gamma_{0}$ is a closed subset of $\Gamma,$ $\psi,$ $\psi*andh$ are given functions on $\Gamma_{0}$ , $\Gamma\backslash \Gamma_{0}$ and $R^{1}$ , respectively, and $ u=$ $( u_{1}, u_{2}, \cdots , u_{N})$ is the unit exterior normal to $\Gamma$ .However, generally, solutions of (P) need not be smooth, so we have to construct boundary conditions in a generalized sense.For this purpose, we introduce a continuous linear operator $B$ from the Banach space $E^{p^{\prime}}(\Omega)$ $=$ { $v=(v_{1},$ $v_{2},$ $\cdots$ , $v_{N}$ ) $;v_{k}\in L^{p^{\prime}}(\Omega),$ $k=1,2,$ $\cdots$ , $N$ , div $v\in L^{p^{\prime}}(\Omega)$ } into $W^{-1/p^{\prime},p^{\prime}}(\Gamma)$ ( $=the$ dual space of $W^{1/p^{r},p}(\Gamma)$ ) such that (v_{1}, v_{2}, \cdots , v_{N})$ with $v_{k}\in \mathcal{D}(\overline{\Omega})$ for all $k$ .Then our boundary condition is given by means of the operator $B$ as follows:$\left\{\begin{array}{ll}u|_{\Gamma}=\psi & a. e. on \Gamma_{0},\\Ba(u)=\psi^{*} & h(u|_{\Gamma}) on \Gamma\backslash \Gamma_{0} (in the distribution sense) ,\end{array}\right.$where $a(u)=(A_{1}(x, u, abla u), \cdots , A_{N}(x, u, abla u))$ .One aim of this paper is to show that equation (P) with generalized boundary conditions of the above type is equivalent to the variational inequality of type (V) associated with it.Another aim is to investigate the continuous dependence of solutions of boundary value problems as formulated above on boundary conditions by using results in [19] and [8].\S 1. Preliminaries.Throughout this paper, let $\Omega$ be a bounded domain in $R^{N},$ $N\geqq 2$ , and assume that the boundary $\Gamma$ of $\Omega$ is very regular, that is, it consists of a finite number of $C^{\infty}$ compact $(N-1)$ -dimensional connected manifolds with $\Omega$ lying on one side of $\Gamma$ .In this section, let $ 1<p<\infty$ and $1/P+1/p^{\prime}=1$ .1.1.The Sobolev space $W^{1p}(\Omega)$ and its trace space $W^{1/p^{\prime},p}(\Gamma)$ .Let us consider the Sobolev space $W^{1p}(\Omega)=\{v\in L^{p}(\Omega);\frac{\partial v}{\partial x_{k}}\in L^{p}(\Omega),$ $k=1,2,$ $\cdots$ , $N\}$ and the trace space of $W^{1p}(\Omega)$ $W^{1/p^{\prime},p}(\Gamma)=\{D\in L^{p}(\Gamma);(\partial)_{p}<\infty\}$ , where $(\partial)_{p}=\int_{\Gamma}\int_{\Gamma}\frac{|0(x^{\prime})-\theta(y^{\prime})|^{p}}{|x-y^{\prime}|^{p+N-2}}d\Gamma_{x^{\prime}}d\Gamma_{y^{\prime}}$ , Pseudomonotone operatOrs 123where $d\Gamma_{x^{\prime}}$ and $d\Gamma_{y^{\prime}}$ mean the surface measure.Norms in these Banach spaces are defined by $\Vert v\Vert_{1,p}=\Vert v\Vert_{L^{p(\rho)}}+\sum_{k=1}^{N}\Vert\frac{\partial v}{\partial x_{k}}\Vert_{L^{p(\rho)}}$ and $[t)]_{\iota/p^{\ovalbox{\tt\small REJECT}},p}=\Vert t)\Vert_{Lp(\Gamma)}+(i))_{p}^{1/p}$ , respectively.The space of all $C^{\infty}$ -functions on $R^{N}$ with compact support in $\Omega$ is denoted by $\mathcal{D}(\Omega)$ and the space of the restrictions of all $C^{\infty}$ -functions on $R^{N}$ to $\Omega$ is denoted by $\mathcal{D}(\overline{\Omega})$ .It is well-known that the operator $\gamma$ : $ u\in$ $\mathcal{D}(\overline{\Omega})\rightarrow$ ( $the$ boundary values of u) is a linear and continuous operator fromequipped with the topology of $W^{1,p}(\Omega)$ into $W^{1/p^{\prime},p}(\Gamma)$ .Since $\mathcal{D}(\overline{\Omega})$ is dense in $W^{1p}(\Omega)$ , there is a unique continuous extension of $\gamma$ to all of $W^{1p}(\Omega)$ .This extension is also denoted by $\gamma$ .Then we know that the range of $\gamma$ is all of $W^{1/p^{\prime},p}(\Gamma)$ and there are positive constants $\lambda_{1}$ and $\lambda_{2}$ such that for all $i)\in W^{1/p^{\prime},p}(\Gamma)$ $inf\{\Vert v\Vert_{1,p} ; v\in W^{1,I)}(\Omega), \gamma v=t)\}$(1.1)$inf\{\Vert v\Vert_{1,p} ; v\in W^{1,p}(\Omega), \gamma v=0\}$ .For a detailed discussion on the operator $\gamma$ , see Gagliardo [7] and Lions- Magenes [16].1.2.Equalities and inequalities for functions in $W^{1p}(\Omega)$ and in $W^{1/p^{\prime},p}(\Gamma)$ .We now recall notions of equalities and inequalities for functions in $W^{1p}(\Omega)$ and in $W^{1/p^{\prime},p}(\Gamma)$ (cf.).Let $\Gamma_{0}$ be a compact subset of $\Gamma$ .Then we say that $\psi\in W^{1/p^{t},p}(\Gamma)$ is non-negative on $\Gamma_{0}$ in the sense of $W^{1/p^{r},p}(\Gamma)$ , if there is a sequence $\{\phi_{k}\}$ $\subset \mathcal{D}(\overline{\Omega})$ such that $\gamma\phi_{k}\geqq 0$ on $\Gamma_{0}$ for all $k$ and $\gamma\phi_{k}\rightarrow\psi s$ in $W^{1/p^{\prime},p}(\Gamma)$ as $ k\rightarrow\infty$ , where we mean by $''\rightarrow s$ " the convergence in the strong topology.For two functions $\psi$ and $\eta$ in $W^{1/p^{1},p}(\Gamma)$ , we define ' $\psi\geqq\eta$ on $\Gamma_{0}$ in the sense of $W^{1/p^{\prime},p}(\Gamma)$ by $\psi-\eta\geqq 0$ on $\Gamma_{0}$ in the sense of $W^{1/p^{\prime},p}(\Gamma)$ .Next, let $F$ be a compact subset of $\Omega$ and $v\in W^{1,p}(\Omega)$ .We then say that $v=0$ on $F$ in the sense of $W^{1p}(\Omega)$ , if there is a sequence $\{\phi_{k}\}\subset \mathcal{D}(\overline{\Omega})$ such $s$ that $\phi_{k}=0$ on a neighborhood of $F$ for all $k$ and $\phi_{k}\rightarrow v$ in $W^{1p}(\Omega)$ as $ k\rightarrow\infty$ .For two functions $v$ and $w$ in $W^{1,p}(\Omega)$ , we define " $v=w$ on $F$ in the sense of $W^{1,p}(\Omega)$ in a way similar to the above.Note that if $v=w$ on $F$ in the sense of $W^{1p}(\Omega)$ , then $v=wa$ .$e$ .on $F$ and $\frac{\partial v}{\partial x_{k}}=\frac{\partial w}{\partial x_{k}}a$ .$e$ .on $F,$ $k=1,2$ , ... ,