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
(*) $latex div(u) = F .$
Here the function $latex u = (u_1, u_2)$ is a vector valued function on the plane $latex u: mathbb{R}^2 rightarrow mathbb{R}^2$, $latex F$ is a scalar valued function on the plane $latex F: mathbb{R}^2 rightarrow mathbb{R}$, and $latex div$ is the divergence operator
$latex div(u) = frac{partial u_1}{partial x} + frac{partial u_2}{partial y} .$
In its simplest form, the problem is: given a specified function $latex F$, does there exist a solution $latex u$ that satisfies $latex div(u) = F$?
Classically, a solution to equation (*) means a differentiable function $latex u$ (meaning that both $latex u_1$ and $latex u_2$ are differentiable functions) such that
$latex div(u) (x,y)= frac{partial u_1}{partial x}(x,y) + frac{partial u_2}{partial y}(x,y) = F(x,y) $
holds for every $latex (x,y) in mathbb{R}^2$. 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 $latex u$ 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 $latex F in C = C(mathbb{R}^2)$ and that $latex u in C^1= C^1(mathbb{R}^2)$ is a solution to (*). It then follows that for every smooth function $latex w in C_c^infty(mathbb{R}^2)$ (i.e., $latex w$ is an infinitely differentiable compactly supported function; sometimes such functions $latex w$ are called test functions) the following holds:
(**)$latex int (frac{partial u_1}{partial x}(x,y) + frac{partial u_2}{partial y}(x,y)) w(x,y) dx dy = int F(x,y) w(x,y) dx dy .$
In fact, if $latex F in C$, then $latex u in C^1$ is a classical solution to (*) if and only if the above equality of integrals holds for every $latex w in C_c^infty$. This follows from the following exercise.
Exercise A: A function $latex f in C(mathbb{R}^n)$ is everywhere zero if and only if for all $latex w in C_c^infty(mathbb{R}^n)$, $latex int f w = 0$.
If we integrate (**) by parts, we find that $latex u$ is a classical solution to (*) if and only if
$latex -int (u_1 w_x + u_2 w_y) = int F w$
or
(*’) $latex int u cdot grad(w) = -int F w ,$
for all $latex w in C_c^infty$. So (*) is equivalent to (*’) for $latex u in C^1$ and $latex F in C$ (here $latex grad(w) = (w_x, w_y)$ is the gradient of $latex w$). But (*’) makes sense also if $latex u$ and $latex F$ are merely locally integrable. Thus for a locally integrable $latex F$, we say that a locally integrable function $latex u$ 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 $latex F$ belongs to a certain class of functions, and ask whether a solution $latex u$ in a given class of functions exists. We will now show that for every $latex F in L^2$ there exists an $latex F in L^infty$ such that $latex u$ is weak solution to $latex div(u) = F$.
There are other notions of generalized solutions, see also Terry Tao’s PCM article or the Wikipedia article.
2. The existence of $latex L^infty$ solutions to $latex div(u) = F$
Let us fix some notation. For simiplicity, let all our functions be real valued. We let $latex L^1 oplus L^1$ denote the space of all pairs $latex (f,g)$, where $latex f,g in L^1(mathbb{R}^2)$. We equip this space with the norm
$latex |(f,g)| = |f|_1 + |g|_1.$
Likewise, $latex L^infty oplus L^infty$ is the space of pairs of functions with the norm
$latex |(f,g)| = max{|f|_infty, |g|_infty }.$
Exercise B: $latex L^1 oplus L^1$ is a Banach space, and $latex (L^1 oplus L^1)^* = L^infty oplus L^infty$.
Theorem 1: For every $latex F in L^2$, there exists a $latex u = (u_1, u_1) in L^infty oplus L^infty$ such that solves (in the weak sense) the PDE $latex div(u) = F$.
Proof: Let $latex M subset L^1 oplus L^1$ be the space
$latex M ={(f,g) in L^1 oplus L^1 : exists w in C^infty_c . (f,g) = grad(w) }.$
Since $latex M$ is the range of a linear map, it is a linear subspsace of $latex L^1 oplus L^1$. The following exercise is not difficult.
Lemma 2: If $latex (f,g) in M$, then there is a unique $latex w in C_c^infty$ for which $latex (f,g) = grad(w)$. The map $latex (f,g) mapsto w$ is linear and bounded as a map from $latex M subset L^1 oplus L^1$ into $latex L^2$.
Assume the lemma for now, and let us proceed with the proof of the theorem. On $latex M$ we define the linear functional
$latex phi : M rightarrow mathbb{R}$
by
$latex phi(f,g) = – int F w $,
where $latex w in C^infty_c$ is such that $latex (f,g) = grad(w)$. Now since $latex F in L^2$ by assumption, the map $latex w mapsto – int Fw $ is a bounded functional on $latex L^2$. Using this fact together with Lemma 2 we conclude that $latex phi$ (which is nothing but the composition of the map $latex M ni grad(w) mapsto w in L^2$ with the map $latex L^2 ni w mapsto – int Fw$) is a well defined, linear and bounded functional on $latex M subset L^1 oplus L^1$. By the Hahn Banach extension theorem (Theorem 12 in Notes 6), $latex phi$ extends to a bounded functional $latex Phi$ on $latex L^1 oplus L^1$. By Exercise B, there exists a $latex u in L^infty oplus L^infty$ such that $latex Phi(f,g) = int (u_1 f + u_2 g) $ for all $latex f, g in L^1 oplus L^1$. Restricting only to elements of the form $latex (f,g) = grad(w) in M$, we find that
$latex int u cdot grad(w) = phi(grad(w)) = -int F w$
for all $latex w in C_c^infty$. In other words, $latex u in L^infty oplus L^infty$ is a weak solution to the equation $latex div(u) = F$.
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.
Proof of Lemma 2: Since the gradient operator $latex grad : C^infty rightarrow C^infty oplus C^infty$ annihilates only constant functions, its restriction to $latex C_c^infty$ has no kernel. Therefore, the linear transformation $latex grad : C_c^infty rightarrow M$ has a linear inverse $latex grad^{-1}: M rightarrow C^infty_c$ which sends every $latex (f,g) in M$ to the unique $latex w in C_c^infty$ such that $latex grad(w) = (f,g)$. The only nontrivial issue is boundedness with respect to the appropriate norms.
The operator $latex grad^{-1}$ actually has a nice formula
$latex grad^{-1}(f,g)(x,y) = int_{-infty}^x f(t,y) dt .$
Thus, if $latex w in C^infty_c$, we have
$latex w(x,y) = int_{-infty}^x frac{d w}{d x}(t,y) dt .$
We obtain the estimate $latex |w(x,y)| leq int_{-infty}^infty |frac{d w}{d x}(t,y)| dt$. Similarly, $latex |w(x,y)| leq int_{-infty}^infty |frac{d w}{d y}(x,s)| ds$. Multiplying the two estimates that we obtained we have
$latex |w(x,y)|^2 leq int_{-infty}^infty |frac{d w}{d x}(t,y)| dt times int_{-infty}^infty |frac{d w}{d y}(x,s)| ds .$
Integrating with respect to $latex x$ and $latex y$, we obtain $latex |w|_2^2 leq |frac{dw}{dx}|_1 |frac{dw}{dy}|_1 leq frac{1}{2}|grad(w) |^2_{L^1 oplus L^1}$, as required.