Does "solutions of an $n$-th order ODE form an $n$-dimensional vector space" somehow generalise to PDEs?

Question

It is well known that the set of solutions $u:\mathbb{R}\rightarrow \mathbb{R}$ of an $n$-th order, linear, homogeneous ordinary differential equation $$a_n(x)\frac{d^n u}{dx^n}+\dots + a_1(x)\frac{du}{dx}+a_0(x)u=0$$ form an $n$-dimensional vectorspace under the usual addition and scalar multiplication of functions. Does this generalise to PDEs, and if so how? For example, the PDE $$\frac{\partial^2u}{\partial x^2}=0$$ has solutions $u=f(y)x+g(x)$ for any functions $f,g$. Clearly the set of solutions has infinite dimension when simply viewed, as in the ODE case, as a vector space under the usual addition and scalar multiplication of functions. However, it is parametrised by two functions. Is there some meaningful sense in which the solution space is $2$-dimensional? And if so, is it true that any set of solutions $u:\mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}$ to a second-order, linear, homogeneous PDE $$a(x,y)\frac{\partial^2 u}{\partial x^2} + b(x,y)\frac{\partial^{2} u}{\partial x\partial y} + c(x,y)\frac{\partial^2 u}{\partial y^2} +d(x,y)\frac{\partial u}{\partial x} + e(x,y)\frac{\partial u}{\partial y} + a_1(x)\frac{\partial u}{\partial x} + a_0(x,y)u=0$$ is $2$-dimensional in that sense?

Answer 1

In many ways the natural generalization of ODEs are hyperbolic PDEs (in that they admit wellposed initial value problems). What you have here is one of those ways. Roughly speaking for a linear hyperbolic PDE on $\mathbb{R}^{1+n}$ (where $n$ may be zero and in which case we have an ODE), the solution is entirely determined by $k$ free functions prescribed on $\{0\} \times \mathbb{R}^n$, where $k$ is the degree of the PDE. That you have a finite dimensional vector space in the ODE case is just a coincidence due to the triviality of $\mathbb{R}^0 := \{0\}$.

Answer 2

It's not 2-dimensional because, as you observed, the solution contains two arbitrary functions of 1 variable. This means that the solution space has infinite dimension, but the right way to think about "counting" solutions is that the general solution contains a certain number of functions involving a certain number of independent variables. So for the example above, we say that the space of solutions "is parametrized by 2 arbitrary functions of 1 variable." Intuitively, this means that in order to determine a unique solution, you have to specify initial data along some curve in $\mathbb{R}^2$ consisting of two functions along the curve. For your example above, typical initial data would take the form

$$ u(y,0) = g(y), \qquad u_x(y,0) = f(y). $$

Then the corresponding solution would be $$ u(x,y) = x f(y) + g(y). $$

For a general PDE, the story is a little more complicated - for instance, there may or may not exist global solutions for a given initial value problem, and even for local solutions the question of how an initial value problem should be posed (e.g., along which curves in $\mathbb{R}^2$ might it be appropriate to specify initial data?) depends on the PDE---mainly on the configuration of its characteristic curves. But most of the time (and certainly in the linear case) the space of local solutions to a single nondegenerate second-order PDE in a neighborhood of some point $(x,y) \in \mathbb{R}^2$ will be parametrized by 2 arbitrary functions of 1 variable.

Answer 3

If coefficients are real analytic, then real analytic solutions can be parametrized locally by their Cauchy data on a non-characteristic hypersurface. This is the Cauchy-Kowalevsky Theorem. However this parametrization is continuous for interesting topologies (essentially) only when the differential operator is hyperbolic with respect to the "initial" surface; see Willie Wong's answer. This excludes - except in the ODE case - the Laplace operator. The solutions of the homogeneous Laplace equation (and of many other elliptic equations) can however be parametrized by a single function, namely the trace on the boundary of a domain (bounded and with sufficiently regular boundary); this is the solution theory of the Dirichlet problem.

Other kinds of parametrization of the general solution of a PDE are given (when coefficients are constant, or on a symmetric space) by the Fourier transform (e.g. the Ehrenpreis' Fundamental Principle) or, e.g. for wave equations, the Radon transform. Here, of course, finite sums are replaced by integrals; so there is no meaningful parameter count. I mention this because I think that, for PDE theory, a focus on parametrization by finitely many parameters is too narrow.