We consider a semicoercive variational inequality (V) (see def. below) under nonlinear mixed boundary conditions: u≥o on r1and u≤o on r2. Here r1and r2are the basic components of the boundary ∂Ω of a bounded domain Ω ⊂ ℝ2. Problem (V) corresponds to a boundary value problem for the Poisson equation Δu=f in Ω, as well as to a convex minimization problem (M). We study the questions of existence, uniqueness and regularity of the solutions to this problem. One of the tools involved is a penalty method which could also be interpreted as a singular perturbation or Yosida-approximation technique. The paper extends the results of [9] to noncoercive boundary conditions.