Abstract: We will present a one-dimensional variational model which has been recently introduced in order to study the epitaxially strained growth of quantum dots, or "islands", that is, agglomerates which are formed when letting a thin film of some material on some substrate. This model has been done for the case of Silicium over Germanium, which is a particularly important case (quantum dot lasers and many optical or optoelectronic devices). The energy to be minimised is the sum of two parts, a surface energe and an elastic one. There are three peculiarities of this model, with respect to the similar ones already proposed in the past: first of all, the elastic energy is a concave functional, which gives of course troubles to the existence of minimizers. The second one is that, due to the constraint coming from the crystallin structure of the materials, the "admissible shapes" of these islands are only those for which the slope belongs a.e. to a given finite set of admissible angles, and this of course prevents the compactness of the set of the admissible shapes. Finally, among these admissible angles there is not the horizontal one, thus the islands cannot start nor end tangentially to the substrate: this phenomenon, usually called "mismatch", is very important for the applications (e.g., for all the observed cases where the islands are not symmetric), but it prevents the well known result of the "zero contact angle" which usually holds in this kind of models. In our talk, after having described the general model and its main characteristics, we will present the relaxation result and we will show some first important properties of the minimizers, which were already been experimentally and numerically observed, but of which a mathematical proof was still missing. We will also describe the other main properties which have been observed, and have yet not been formally proved. This is a joint work with I. Fonseca and B. Zwicknagel.