We consider variations of the low temperature chemical potential and magnetization of laterally confined two-dimensional interacting electrons in a strong magnetic field. Cases of axially symmetrical (a quantum dot) and strongly elongated (a quantum wire) spatial distributions of electron density are studied. We commonly refer to both types of systems as ``islands.'' The calculations are performed in the mean-field approximation. In this approximation the system consists of alternating regions of compressible and incompressible liquids. We are focusing on islands with typical widths b, which are much larger than the Bohr radius in a semiconductor ${\mathit{a}}_{\mathit{B}}$, and argue that the density profile is formed mainly by electrostatic forces, has a domelike shape, and changes only slightly in the whole range of magnetic fields. We calculated numerically the chemical potential and the magnetization as functions of the magnetic field and found them to be in sharp contrast to the conventional result for noninteracting electrons. We suggest a simple analytic theory adequately explaining these functional dependences. The results are compared with the experimental data.