The jellium model is commonly used in condensed matter physics to study the properties of a two-dimensional electron gas system. Within this approximation, one assumes that electrons move in the presence of a neutralizing background consisting of uniformly spread positive charge. When properties of bulk systems (of infinite size) are studied, shape of the jellium domain is irrelevant. However, the same cannot be said when one is dealing with finite systems of electrons confined in a finite two-dimensional region of space. In such a case, geometry and shape of the jellium background play a role on the overall properties of the system. In this work, we assume that the region where the electrons are confined is represented by a jellium background charge with an elliptical shape. It is shown that, in this case, the Coulomb self-energy of the elliptically shaped region can be exactly calculated in closed analytical form by using suitable mathematical transformations. The results obtained reveal the external influence of geometry/shape on the properties of two-dimensional systems of few electrons confined to a small finite region of space. The two-dimensional electron gas (2DEG) model has received a great deal of attention in condensed matter physics due to the richness and complexity of the emerging phenomena associated with it. The variety of possible scenarios makes this model fascinating from a theoretical and experimental point of view. In particular, a 2DEG in a strong perpendicular magnetic field has come to the forefront of current research as a result of the discovery of the integer quantum Hall effect (IQHE) However, there have been recent developments in the field of nanotechnology that make possible the fabrication of finite systems of few electrons confined in a finite 2D domai
