On the existence of generalized solutions with finite Dirichlet integral for basic exterior boundary value problems (Q1814317)

The author considers the Dirichlet and Neumann problems for the elliptic equation \[ (a^{ij}(z)u_{z_ j})_{z_ i}=f(z), z\in\Omega', \] where \(z=(x,y),\;\Omega'=\omega\times\mathbb{R}^ k\) and \(\omega\subset\mathbb{R}^ n\) is a domain with bounded complement. It is assumed that \(a^{ij}\) and \(f\) are 1-periodic with respect to the components of \(y\in\mathbb{R}^ k\). The finiteness of the Dirichlet integral of a solution is examined.