Free Boundary Problem refers to an a priori unknown interface, along which a possible phase transition or a qualitative change in the given equations occurs. The subject area has developed in the last 50 years, and has found new branches of directions.

Recently several new research directions have arrived, with many new and challenging problems.

Here I shall mention four main problems that might be of interest as well as viable:

i) Vectorial free boundary problems.

ii) Vectorial free transmission problems.

iii) Free boundaries with higher order degeneracies.

Iv) Symmetry problems for vectorial free boundaries.

One important aim is to bridge the regularity theory of the free boundaries and transmission problems, whenever appropriate and possible. A second, and equality important, problem is to develop new tools for treating these problems in the case of coupled linear and ultimately nonlinear systems of differential equations, that has so far been almost untouched. A third problem is the development of the theory for the free boundaries of obstacle type with higher degeneracies that has recently arisen in scattering theory. In all the above problems, main questions concern regularity of solutions, and the free boundaries they give rise to. Finally the symmetry of free boundary problems for systems is also an important area that is under development.

Several of these directions might be interesting to work on for early career persons, and Ph.D. students. Combinations of these problem may also be possible.

It is noteworthy that knowledge of programming and numerical skills is usually highly important (but not necessary) ti understand any planned projects by playing with simple numerical examples. Indeed, some important steps in the process of studying these problems would be to use numerics to indicate that the suggested problem is somehow correct.

You may read about the free boundaries at articles in the references below.
Also several open problems are already published at Scilag-page, that you con fined them, by search with topics.

Article
Regularity issues for semilinear PDE-s (a narrative approach)

If you plan to formulate more than one problem all sharing the same background (e.g. they are all from the same paper) then please choose "Group", otherwise select "Single" option.