On the Dirichlet problem for a semi-linear elliptic equation with \(L^ 2\)-boundary data
The aim of this paper is to investigate the existence of a solution of the Dirichlet problem for a semilinear equation of elliptic type \[ -\sum^{n}_{i,j=1}D_ i(a_{ij}(x)D_ ju)=f(x,u,Du)\quad in\quad Q;\quad u(x)=\phi(x)\quad on\quad \partial Q, \] where \(\phi \in L^ 2(\partial Q)\) and is not, in general, a trace of a function belonging to \(W^{1,2}(Q)\). Here Q denotes a bounded region in \(R_ n\) with \(\partial Q\subset C^ 2\). The boundary condition is understood in the sense of \(L^ 2\)-convergence, i.e., \[ \lim_{\sigma \to 0}\int_{\partial Q}[u(x_{\delta}(x))-\phi(x)]^ 2dS_ x=0, \] where a \(C^ 1\)-mapping \(x_{\delta}:\partial Q\to \partial Q_{\delta}\) is defined by the formula \(x_{\delta}(x)=x-\delta \nu(x),\quad | x_{\delta}(x)-x| =\delta,\) where \(\nu\) (x) is the outward normal to \(\partial Q\) at x and \(Q_{\delta}=Q\cap \{\min_{y\in \partial Q}| x-y|>\delta \}.\) Under various assumptions on a nonlinearity f the existence of a solution is established in the space \(\tilde W^{1,2}(Q)=\{u:\quad u\in L^ 2(Q),\quad Du\in L^ 2_{loc}(Q)\) and \[ \int_{Q}| Du(x)|^ 2r(x)dx+\sup_{0<\delta \leq d}\int_{\partial Q_{\delta}}u(x)^ 2dS_ x<\infty \}, \] where \(r(x)=dist(x,\partial Q)\).
scientific article; zbMATH DE number 3855585