**Capillary equations and related topics **

In fluid mechanics capillarity is the ability of a liquid to flow in narrow spaces without the assistance of, or in opposition to, external forces. It occurs because of inter-molecular attractive forces between the liquid and the solid surrounding surfaces. More in general, capillarity is the set of phenomena due to interactions between the molecules of a fluid and of another material, which can be a solid, a liquid or a gas, on their separation surface (interface), said capillarity surface. With the advent of miniaturized technology, also systems combining the effects of electrostatic and capillary forces are receiving more and more attention, in order to fully understand how these electro-capillary systems operate at small scales. In very recent years some new models have been introduced with the aim of taking into account of the full effects of capillarity.

From the mathematical point of view these phenomena are governed by highly nonlinear partial differential equations, which exhibit various peculiarities: the possible lack of existence or uniqueness of solutions, the general lack of regularity, the possible development of singularities either in the interior or at the boundary. The study of these equations, which involve the mean curvature operator, is a classical subject, but still of great actuality, due to the large number of open problems, having a great interest even from the point of view of the applications.

Their analysis requires the use of sophisticated tools of advanced mathematics: geometric measure theory, variational methods in spaces of functions of bounded variation, non-smooth analysis techniques, combined with other typical techniques of nonlinear analysis, such as critical points theory, topological degree, bifurcation techniques. Proposed research projects include, but are not limited to, the study of:

a) existence, non-existence, multiplicity, and qualitative properties (such as regularity and stability) of solutions of the capillarity equations, on bounded or unbounded domains, which appear as sub-critical points (not necessarily minimizers) of the functional representing the mechanical energy of the system, with special attention to superlinear/sublinear indefinite problems.

b) micro-electro-capillary systems, which lead to a class of partial differential equations involving the mean curvature operator with a singular perturbation, whose study has been, for the moment, confined to low dimensional problems, possibly with symmetries. A challenging problem is to build a mathematical theory (existence, multiplicity and qualitative properties) of these anisotropic mean curvature equations governing electro-capillary systems in the general N-dimensional case.