**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.