Using Functional Analysis for Differential Equations

I am a little surprised you haven't gotten more responses here; perhaps it is because once you see the functional analytic perspective on PDEs, it becomes hard to think about PDEs without functional analysis.

Anyway here is a sketch of an example. Depending on how much functional analysis you've seen, it might not totally make sense; but I wanted to keep the example short to preserve the high level point. Fix a domain $\Omega$ and consider the Laplace's equation $\Delta u = 0$ on $\Omega$ with Dirichlet boundary conditions. Multiplying by a test function $v$ and integrating by parts, we see that $$-\int_{\Omega}v\Delta u=\int_{\Omega}\nabla u\cdot\nabla v-\int_{\partial\Omega}\frac{\partial u}{\partial \nu}vd\mathcal{H}^{d-1}.$$ If we stipulate that the test function $v$ is also zero on the boundary of $\Omega$, this implies that $$\int_\Omega \nabla u \cdot \nabla v = 0.$$ We say that this is the "weak" form of Laplace's equation.

Here is where the functional analysis perspective comes in handy. The "natural" function space here is the Sobolev space $H_0^1 (\Omega)$. On this space, it turns out[1] that $\int_\Omega \nabla u \cdot \nabla v$ is an equivalent inner product. Therefore, we can rewrite the weak form of Laplace's equation as $\langle u, v \rangle_{H_0^1 (\Omega)} = 0$. Now, we allow $v$ to range over the entire space $H_0^1 (\Omega)$. Defining the continuous linear functional $L$ on $H_0^1(\Omega)$ by $L(v)=0$, we can further rewrite the weak form of Laplace's equation as $$\forall v \in H_0^1 (\Omega) \quad \langle u, v \rangle_{H_0^1 (\Omega)} = L(v).$$ In other words, there is a solution to the weak form of Laplace's equation if there is a $u$ which solves the problem above. But because $H_0^1(\Omega)$ is a Hilbert space, this is automatically true thanks to the Riesz representation theorem (and moreover, the $u$ is unique!). That is, there is a unique $u\in H_0^1 (\Omega)$ such that $\forall v \in H_0^1 (\Omega)$, $\int_\Omega \nabla u \cdot \nabla v = 0.$

In particular, since any "bona fide" solution $u$ to $\Delta u = 0$ automatically satisfies the weak Laplace equation, this implies that there is at most one solution to $\Delta u = 0$.

In summary: we have taken a fairly standard PDE, transformed it slightly into a "weak" analogue in the "right" space, and then shown that solving the weak analogue is just a direct application of some functional analysis result.

[1] See here: Norm and scalar product of $H_0^1(\Omega)$

There are in fact many applications of functional analysis and functional analytic techniques to the theory of differential equations, both ordinary and partial. Indeed, so many such applications exist that it is impossible to even begin to provide a comprehensive list in such a small space. Therefore I will focus on two of most central, one in each of the fields ordinary and partial differential equations.

To begin with an example of how functional analytic techniques come to play in the theory of ordinary differential equations, we may turn to the Picard-Lindeloef theorem, which is the central result affirming the existence and uniqueness of solutions to a large and important class of ODEs, namely those of the form

$\dot{\vec y} = f(\vec y, t), \tag 1$

where $f(\vec y, t)$ is assumed to be Lipschitz continuous in $\vec y$ and jointly continuous in $\vec y$ and $t$. Picard-Lindeloef is typically proved via an invocation of the Banach fixed point theorem, also sometimes known as the contraction mapping theorem, which I think may be properly regarded as within the domain of functional analysis per se, although some may disagree. Allow me to explain:

A major program of functional analysis is to study objects which are infinite-dimensional generalizations of their finite dimensional counterparts; thus we work with normed, Banach and Hilbert spaces etc., and general linear operators between them in lieu of more specialized functions such as derivatives and integrals and so forth. Of course, we still draw upon these operations from ordinary calculus, or even real and complex analysis, in order to map out just where a fruitful application of a more functional analytic approach might be found. It strikes me that a fruitful basic example here is the isometric isomorphism between $L^2(\Bbb R)$ and itself provided by the Fourier transform; here we see both a very specific integral operator pf the form

$\mathcal F(f)(\omega) = \displaystyle \int_{-\infty}^\infty f(x) e^{-i\omega}x \; dx, \; f(x) \in L^2(\Bbb R) \tag 2$

as well as an isometric linear map. The Fourier transform in some sense lives at the boundary of real and complex analysis and the theory of operators on Hilbert spaces; many aspects of functional analysis dwell at similar boundaries between otherwise seemingly disparate subjects; indeed, it is the functional analytic approach which unifies such differing approaches to certain problems.

So how does the Banach fixed point theorem fit into the general program of functional analysis? Well, for one thing, it pertains to functions

$T:X \to X, \tag 3$

where $X$ is a complete meric space, such that there exists some $k$,

$0 < k < 1, \tag 4$

such that

$x_1, x_2 \in X \Longrightarrow d(f(x_1)), f(x_2)) < k d(x_1, x_2), \tag 5$

where

$d:X \times X \to \Bbb R \tag 6$

is the metric on $X$; the theorem then affirms the existence of a unique point

$x^\ast \in X \tag 7$

such that

$\displaystyle \lim_{i \to \infty} f^i(x) = x^\ast \tag 8$

for any $x \in X$. We note that many linear spaces provide us with subspaces such as the complete metric space $X$ (note that we do not stipulate $X$ itself be a linear subspace); therefore it should come as no surprise that the usual linear spaces encountered in analysis often contain (not necessarily linear) subpaces $X$ and maps $T:X \to X$ to which the fixed point theorem applies.

In fact, perhaps the premiere application of the Banach fixed point theorem in the field of ordinary differential equations is to the proof of the above-mention Picard-Lindeloef theorem; in this case, assuming

$\dim \vec y = n, \tag 8$

and a solution to (1) is sought on some interval

$[a, b] \subset \Bbb R, \tag 9$

with the specified initial condition

$\vec y(a) = \vec y_0, \tag{10}$,

we look at the set

$X = \{ \vec y(t) \in C^0([a, b], \Bbb R^n, \; \vec y(a) = \vec y_0 \}, \tag{11}$

and define

$T: X \to X \tag{12}$

via

$T(\vec y(t)) = \vec y_0 + \displaystyle \int_a^t f(\vec y(s), s) \; ds; \tag{13}$

for

$\vec y_1(t), \vec y_2(t) \in X; \tag{14}$

we have, assuming $k$ is a Lipschitz constant for $f$, that is

$\Vert f(\vec y_1, t) - f(\vec y_2, t) \Vert \le k \Vert y_1 - y_2 \Vert, \tag{14.5}$

$\Vert T(y_2(t)) - T(y_1(t)) \Vert = \displaystyle \sup_{t \in [a, b]} \vert T(y_2(t)) - T(y_1(t)) \vert$ $= \displaystyle \sup_{t \in [a, b]} \left \vert \displaystyle \int_a^t (f(\vec y_2(s), s) - f(\vec y_1(s), s)) \; ds \right \vert \le \displaystyle \sup_{t \in [a, b]} \int_a^t \vert f(\vec y_2(s), s) - f(\vec y_1(s), s) \vert \; ds$ $= \displaystyle \int_a^b \vert f(\vec y_2(s), s) - f(\vec y_1(s), s) \vert \; ds \le \displaystyle \int_a^b k \vert y_2(s) - y_1(s) \vert \; ds$ $\le k \displaystyle \int_a^b \sup_{t \in [a, b]} \vert \vec y_2(t) - \vec y_1(t) \vert \; ds = k \displaystyle \int_a^b \Vert \vec y_2(t) - \vec y_1(t) \Vert \; ds = k(b - a)\Vert \vec y_2(t) - \vec y_1(t) \Vert \; \tag{15}$

this shows that $T$ is a contraction mapping if

$k(b - a) < 1, \tag{16}$

and hence under this condition (13) has a unique fixed point such that

$\vec y(t) = \vec y_0 + \displaystyle \int_a^t f(\vec y(s), s) \; ds; \tag{17}$

upon differentiating this equation we obtain

$\dot{\vec y(t)} = f(\vec y(t), t), \tag{18}$

showing such $\vec y(t)$ is the unique solution to (1).

Thus we see how functional analytic techniques come to play in the theory of ordinary differential equations.

Applications to partial differential equations are perhaps even more obvious. I think here, for example, of the Babuška–Lax–Milgram_theorem and its relation to weak solutions of certain PDEs, as is outlined in the answer of pseudocydonia. Since this answer is already perhaps overlong, I leave it to the reader to invesitgate the link I have provided to Babuška–Lax–Milgram for a more complete picture of this application of functional analysis to PDEs.