Advanced Analysis, Notes 8: Banach spaces (application: weak solutions to PDEs)
by Orr Shalit
Today I will show you an application of the Hahn-Banach Theorem to partial differential equations (PDEs). I learned this application in a seminar in functional analysis, run by Haim Brezis, that I was fortunate to attend in the spring of 2008 at the Technion.
As often happens with serious applications of functional analysis, there is some preparatory material to go over, namely, weak solutions to PDEs.
1. Weak solutions to PDEs
In our university, Ben-Gurion University of the Negev, Pure Math majors can finish their studies without taking a single course in physics. Therefore I will say the obvious: partial differential equations are one of the most important and useful branches of mathematics. It is a huge subject. When working in PDEs one requires an arsenal of different tools, and functional analysis is just one of the many tools that PDE specialists use.
Since my only goal here is to give an example, and so as to be very very concrete, I will discuss only one PDE, the PDE
Here the function is a vector valued function on the plane , is a scalar valued function on the plane , and is the divergence operator
In its simplest form, the problem is: given a specified function , does there exist a solution that satisfies ?
Classically, a solution to equation (*) means a differentiable function (meaning that both and are differentiable functions) such that
holds for every . The question whether a classical solution exists or not is a respectable mathematical question, but as I noted above, PDEs arise in applications and exist for applications, and it is sometimes not reasonable to expect that the solution will be differentiable or even continuous. So one is led to consider weak solutions, that is, functions which are not differentiable, but which solve the PDE (*) in some sense.
(There is another reason to consider weak solutions besides the need the arises in applications: sometimes the existence of a classical solutions is shown in two steps. First step: a weak solution is shown to exist. Second step: the weak solution is shown to enjoy some regularity properties and is shown to be a solution in the classical case).
In what sense? Assume that and that is a solution to (*). It then follows that for every smooth function (i.e., is an infinitely differentiable compactly supported function; sometimes such functions are called test functions) the following holds:
In fact, if , then is a classical solution to (*) if and only if the above equality of integrals holds for every . This follows from the following exercise.
Exercise A: A function is everywhere zero if and only if for all , .
If we integrate (**) by parts, we find that is a classical solution to (*) if and only if
for all . So (*) is equivalent to (*’) for and (here is the gradient of ). But (*’) makes sense also if and are merely locally integrable. Thus for a locally integrable , we say that a locally integrable function is a weak solution to (*) if it satisfies (*’). Experience has shown that this is a reasonable notion of solution to the original PDE.
Now we are free to study (*’) where belongs to a certain class of functions, and ask whether a solution in a given class of functions exists. We will now show that for every there exists an such that is weak solution to .
2. The existence of solutions to
Let us fix some notation. For simiplicity, let all our functions be real valued. We let denote the space of all pairs , where . We equip this space with the norm
Likewise, is the space of pairs of functions with the norm
Exercise B: is a Banach space, and .
Theorem 1: For every , there exists a such that solves (in the weak sense) the PDE .
Proof: Let be the space
Since is the range of a linear map, it is a linear subspsace of . The following exercise is not difficult.
Lemma 2: If , then there is a unique for which . The map is linear and bounded as a map from into .
Assume the lemma for now, and let us proceed with the proof of the theorem. On we define the linear functional
where is such that . Now since by assumption, the map is a bounded functional on . Using this fact together with Lemma 2 we conclude that (which is nothing but the composition of the map with the map ) is a well defined, linear and bounded functional on . By the Hahn Banach extension theorem (Theorem 12 in Notes 6), extends to a bounded functional on . By Exercise B, there exists a such that for all . Restricting only to elements of the form , we find that
for all . In other words, is a weak solution to the equation .
This may seem a little magical, but don’t forget that we still haven’t proved Lemma 2. Lemma 2 is a typical example of an estimate that one has to prove in order to apply functional analysis to PDEs, and falls under the wide umbrella of the Sobolev–Nirenberg inequalities.
Note: I actually proved Lemma 2 in class (it’s important!!) but I don’t have time right now to do it here. Students who want to see the proof should talk to me.
Proof of Lemma 2: Since the gradient operator annihilates only constant functions, its restriction to has no kernel. Therefore, the linear transformation has a linear inverse which sends every to the unique such that . The only nontrivial issue is boundedness with respect to the appropriate norms.
The operator actually has a nice formula
Thus, if , we have
We obtain the estimate . Similarly, . Multiplying the two estimates that we obtained we have
Integrating with respect to and , we obtain , as required.