• Bernoulli problem
• Free boundary problem
• stability

The Bernoulli problem has been well studied since Alt-Caffarelli's s paper in '81.
It consists in a PDE with overdetermined boundary conditions, which makes this a free boundary problem, in the sense that, since not every domain admits a solution, the main questions concern the domains that have a solution.

In this Master Thesis we try to classify solutions that are stable with respect to the Alt-Caffarelli functional in the whole space in low dimensions.
It is easy to build some solutions, but every solution constructed is 1-dimensional, in the sense that it depends only on 1 component, hence it is natural to conjecture that in low dimensions every stable solution is 1-dimensional and we prove some results that go in this direction.

We recall a proof of the conjecture in the case of axially symmetric solutions and the prove in the class of 1-homogeneous stable solutions.
Each one of these proof test the stability inequality with a non-trivial function involving the solution and its derivatives, and we try to adapt the same strategy in the classification of stable solutions in low dimensions.
In particular, we give two different proofs of the conjecture in dimension 2, while we prove a partial result in dimension 3, namely we prove the conjecture under additional metric assumptions on the free boundary. This result proves for instance that if a solution has a zero set with convex connected components, or has the free boundary that is a minimal surface, then it is 1-dimensional.
Finally, we try to understand better a critical case for the proof of the main result, leading to the study and the proof of instability of an explicit example in dimension 3.