Dirichlet problem for a linear elliptic equation in unbounded domains with \(L^ 2\)-boundary data
The purpose of the paper is to investigate the solvability in \(W^{1,2}_{loc}(\Omega)\) of the Dirichlet problem \(Lu=f\) in \(\Omega\), \(u|_{\partial \Omega}=\phi\), where L is a linear uniformly elliptic operator of order two, \(\Omega\) is the half-space \(\{x=(x_ 1,...,x_ n)\in {\mathbb{R}}^ n| x_ n>0\}\) or the complement to \({\mathbb{R}}^ n\) of a bounded closed set, f is measurable and bounded on \(\Omega\), \(\phi\in\) \(L^ 2(\partial \Omega)\). The boundary \(\partial \Omega\) is assumed to be of class \(C^ 2\) and the boundary condition \(u|_{\partial \Omega}=\phi\) means that \[ \lim_{\delta \to 0}\int_{\partial \Omega}[u(x-\delta \nu (x))-\phi (x)]^ 2dx=0, \] where \(\nu\) (x) is the outward normal to \(\partial \Omega\) at x.
scientific article; zbMATH DE number 3888156
scientific article