OpenYear of origin: 2012
Posted online: 2022-01-25 11:17:43Z by Henrik Shahgholian
Cite as: P-220125.2
This is a previous version of the post. You can go to the current version.
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.
An example of Problem i) is the consideration of the minimisers of \begin{equation} J(\bf u) = \int_{B_1} F(x, \bf u, \nabla \bf u) + G(x, \bf u) \, dx, \end{equation} where one may expect to prove that in the case $G(x,\bf u)$ is Lipschitz in both variables, minimisers have quadratic growth from the FB , and are smooth in a universal vicinity of the FB. Despite the fact that such statements do not hold for classical PDEs, it might be true here, due to presence of the FB. Alternative to this is to study the FB properties.
The second problem above, has the following equation \begin{equation} \hbox{div} ((A + B \chi_D) \nabla \bf u)=0, \qquad \bf u = (u_1, \cdots , u_m), \qquad \hbox{in } B_1 \end{equation} where $D = D(\bf u)$ is a priori unknown domain, with $D \cap B_1 \not = \emptyset$, and $A= A(x,\bf u) $, $B= B(x, \bf u)$ are matrices with positive eigenvalues. Again by imposing certain conditions on $D$ one asks for regularity of solutions as well as the boundary $\partial D$. The conditions being imposed on $D$, parallels the scalar case.
Problem iii) arises in certain applications (both variational and non-variational) the optimal decay rate for solutions at FB points can be higher than quadratic due to the nature of the problem. E.g., the function $u_1 = (x_1)_+^3$ solves $\Delta u_1 = 6x_1 \chi_{\{u_1 > 0\}}$ or $u_2 = x_2(x_1)_+^2$ solves $\Delta u_2 = 2x_2 \chi_{\{u_2 \neq 0\}}$. Such examples easily arise as blow-up (global) solutions to the obstacle problem with higher degeneracies, that is a consequence of the fact that the right hand side $f$ of the equation (before a blow-up) \begin{equation} \Delta u = f(x) \chi_{\{u \neq 0\}} \end{equation} may vanish of some order at a FB point $ x^0 \in \partial \{ u \neq 0\}$.
Recent developments in scattering theory (specially in connection with non-scattering phenomenon) shows the appearance of exactly such type of FB problems. This, and similar applications, as well as the pure mathematical interest and curiosity can force the advancement of the regularity theory of FB towards studying such problems. This is a completely open research direction from a regularity point of view.
For symmetry problems kindly see: https://www.scilag.net/problem/G-191226.1
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