Mixed problems with non-homogeneous boundary conditions in Lipschitz domains for second-order elliptic equations with a parameter

For a second-order elliptic equation involving a parameter, with principal part in divergence form in Lipschitz domain $\Omega$ mixed problems (of Zaremba type) with non-homogeneous boundary conditions are considered for generalized functions in $W^1_2(\Omega )$. The Poincaré–Steklov operators on Lipschitz piece $\gamma$ of the domain's boundary $\Gamma$ corresponding to homogeneous mixed boundary conditions on $\Gamma \setminus \gamma$ are studied. For a homogeneous equation with separation of variables in a tube domain with Lipschitz section, the Fourier method is substantiated for homogeneous mixed boundary conditions on the lateral surface and non-homogeneous conditions on the ends.