# What is a Free Boundary Problem?

### First, What Is A PDE?

We need to start with what is a partial differential equation (PDE). The aim of this page is not to offer a comprehensive introduction to PDEs, so I’ll illustrate the idea with a specific PDE, the well known Laplacian:

$\Delta u(x,y) = \frac{\partial^2 u}{\partial x^2} +\frac{\partial^2 u}{\partial y^2} =0.$

So a function $u(x,y)$ function solves $\Delta u =0$ if the sum of its pure second partials is zero. This is called a classical solution, in contrast to the various weak concepts of a solutions that are encountered in more advanced work.

So a typical PDE problem is to try and find a function $u(x,y)$ that solves the PDE in the interior of a domain $\Omega$ with $u(x,y)$ equal to some given function on the boundary of $\Omega$. A domain is just a connected open subset, in this case of  $\mathbb{R}^2$. This problem is called the Dirichlet problem. And maybe it has a solution, maybe it doesn’t. In this case existence of a solution depends on the domain $\Omega$.

This is called a boundary value problem since, in addition to the PDE that $u$ has to solve, it also has to match some given boundary data on the boundary of the domain. Absent this second condition the problem isn’t meaningful  since there are infinitely many solutions to $\Delta u=0$ (constant functions and planes are easy examples, but there are many more).

### So What’s a Free Boundary Problem Then?

All this brings me to what a free boundary problem is and how it’s different from the above. In a free boundary problem what is known or given to you is the PDE that $u$ needs to solve, boundary data, and an additional `free boundary condition’, but what is not known is the domain over which the solution should solve the PDE. Let’s give an example: We’ll take the unit ball $B_1 \subset \mathbb{R}^2$  and assign boundary data on this ball that is zero where $y<0$ and positive on the part where $y>0$. What we want is a function $u(x,y)$ defined on $B_1$ and some region $D \subset B_1$ with the properties that

1. $\Delta u(x,t) =0$ inside $D$. Note carefully that we do not require that $\Delta u=0$ in all of $B_1$.
2. $u\geq 0$ in $B_1$ and $u>0$ in $D$.
3. Along the edge of $D$ that is inside of $B_1$, which we’ll call $\Gamma$, we want $u_{\nu} =1$. This is called the ‘free boundary condition’. We call $\Gamma$ the free boundary since it’s not fixed at the outset of the problem.

In fact, this is a famous problem in free boundaries. It is a simple case of the celebrated Alt- Caffarelli-Friedman minimization problem.

The difficulty with this problem is that the domain $D$ over which we need $u(x,y)$ to be harmonic is not known. Now one could just pick a domain $D$ and solve the Dirichlet problem on it, but in general the solution would not satisfy the third condition.

This might suggest that the hardest part of this problem would be showing that a solution even exists. But this is not quite true. It turns out that one can find a ‘weak’ solution to the problem.  However, a weak solution does not necessarily solve the problem in the classical sense, and most of the time we like to have a classical solution if at all possible. This happens a lot in PDEs; we have a ‘weak’ solution and hope that we can prove it is in fact classical (and sometimes it’s not, but that’s OK too).

In our case we notice the boundary condition $u_{\nu} =1$ requires two things to hold for $u$ to be a classical solution. First, we would need the normal $\nu$ to exist at every point on the free boundary $\Gamma$, and we would need the function $u$ to be continuously differentiable up to this boundary.

And that’s where one of the really big questions in free boundaries comes from. We want to know how smooth $\Gamma$ is. If it’s not smooth then there’s no hope of $u$ solving the problem in a classical sense. Typical results along these lines involve assuming that $\Gamma$ has some ‘smoothness’ (I use this in a loose sense, the technical term is regularity) and proving that it is in fact smoother than we assumed. So, for example, some of my work has been to prove that a free boundary which is assumed to be Lipschitz (which is not really ‘smooth’ at all) is in fact $C^{1,\alpha}$. In the literature this is called a ‘Lipschitz implies smooth’ type of result.

There are many other sorts of questions that can be asked about free boundary problems. For example, existence, uniqueness, and regularity of the solution (which is often worse than the regularity of the free boundary) are other questions that are important. I’ve emphasized the regularity of the free boundary since it’s a question that’s more specific to free boundary problems; the other questions appear in all branches of PDEs.