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Motivation for question: I know just a little about the general theory of PDEs. I'm working on a project which happens to need examples of PDEs like Laplace's equation. The next step is to look at higher order PDEs in my investigation. I own a few PDE texts and I had hoped to find further examples, but I've noticed that the bulk of my PDE books concern mainly second order PDEs over various dimensional spaces (for example, $n=2$: $u_{xx}+u_{yy}=0$, or $n=3$: $u_{xx}+u_{yy}=u_{tt}$).

My question is this: do higher order equations (like $u_{xxx}=u_{tt}$, to give a silly example) have a well-developed theory?

In the case of ODEs I'm aware that we can reduce to a system of first order equations thus solutions to the $n^{th}$-order problem are furnished by the first-order theory.

Is there a similar story for PDEs? Or, is it just that higher-order PDEs are less interesting?

Thanks in advance for any insights!

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One is the Biharmonic equation – enzotib Aug 1 '12 at 21:39
Less interesting? This seems like an odd assumption to make. – Qiaochu Yuan Aug 1 '12 at 21:43
In physics, there are solid reasons why PDEs of order $\gt 2$ are not of interest. In classical mechanics, you only need two pieces of initial data, the position and velocity, to predict the motion. In quantum mechanics, higher-derivative theories have physically unacceptable behavior:… – Ben Crowell Aug 2 '12 at 1:02
Well, less interesting in the sense that people (on the average) are not interested in it. Admittedly this is not a wise metric with which to judge. – James S. Cook Aug 2 '12 at 1:02
up vote 5 down vote accepted

The first example that comes to mind is the clamped plate equation, which is of 4th order. To see how engineers approach it, click here. More generally, any time your model calls for prescribed values and prescribed derivative on the boundary, you'll probably be dealing with a PDE of order above $2$. For second-order equations such boundary value problems are overdetermined.

There is also a general method for the Dirichlet problem for elliptic operators of any order, namely one can use Gårding's inequality to verify the coercitivity assumption in the Lax-Milgram theorem.

On the other hand, a recent paper by one of my colleagues opens with a sobering statement

There is nothing in the theory of linear strongly elliptic differential operators [of higher orders] that can be called a general existence theorem for solutions to the Neumann problem.

On the positive side, the Fourier transform goes a long way for linear PDE of all orders, especially as far as the regularity of solutions is concerned. The very size of Hörmander's 4-volume treatise "The analysis of linear partial differential operators" suggests that there is something well-developed in there.

One of the things that make second order equations easier to deal with is the fact that the calculus test for maxima/minima is the second derivative test. The maximum principle is a workhorse in the theory of elliptic equations of 2nd order. In contrast, the Hadamard maximum principle for biharmonic equation does not go very far (see page 2 of the paper).

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Sadly my journal access is pretty much just ArXiV in the short term... so I can't see your colleagues' paper. That said, the links are intriguing. I think I can use some of these equations for my project, much appreciated. – James S. Cook Aug 2 '12 at 1:21
@JamesS.Cook It's pretty technical, while you are looking for the wider picture. I just used it for the quote which I happened to remember. – user31373 Aug 2 '12 at 1:24

For some remarks on what the general story looks like you might want to consult the Princeton Companion article about partial differential equations. I don't have much to say myself about this so I'll just mention two examples.

The Korteweg-de Vries equation is of interest to people working in integrable systems (there are other examples at that link). I understand that it has some interesting ties to algebraic geometry.

Another example (really a family of examples) is the following. Let $G$ be a Lie group acting smoothly on a smooth manifold $M$. This induces a map from the Lie algebra $\mathfrak{g}$ to the Lie algebra of smooth vector fields on $M$, which induces an action of the universal enveloping algebra $U(\mathfrak{g})$ of $\mathfrak{g}$ on $C^{\infty}(M)$ by differential operators. Choose $D \in U(\mathfrak{g})$. Then you can write down the differential equation $$Df = 0$$

where $f \in C^{\infty}(M)$. This differential equation should have order the degree of $D$ (with respect to the usual filtration on $U(\mathfrak{g})$). The space of solutions to this differential equation is acted on by the subgroup of $G$ commuting with $D$. In particular, if $D$ lies in the center, the space of solutions is acted on by all of $G$.

For a very simple example, let $G = \mathbb{R}$ act on itself by translation. Then $\mathfrak{g}$ is spanned by ordinary differentiation $D$, the universal enveloping algebra $U(\mathfrak{g})$ is polynomials in $D$, and the corresponding differential equations you can write down are all linear homogeneous ODEs with constant coefficients, all of which admit an action of $G$ (again by translation). Writing down this action explicitly gives generalizations of the angle addition formulas (which you obtain in the special case $D^2 + 1$).

For a non-abelian example, let $G = \text{SO}(3)$ act on $\mathbb{R}^3$ by rotation. Then the image of $\mathfrak{g} = \mathfrak{so}(3)$ in vector fields on $\mathbb{R}^3$ is spanned by

$$L_x = y \frac{\partial}{\partial z} - z \frac{\partial}{\partial y}$$ $$L_y = z \frac{\partial}{\partial x} - x \frac{\partial}{\partial z}$$ $$L_z = x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x}.$$

These vector fields generate rotation about the $x, y, z$-axes respectively. The Casimir operator $$L^2 = L_x^2 + L_y^2 + L_z^2$$

turns out to generate the center of $U(\mathfrak{so}(3))$, so the space of solutions to the differential equation $p(L^2) f = 0$ for any polynomial $p$ is acted on by all of $\text{SO}(3)$ and has order $2 \deg p$.

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This is a very nice answer. Although, I'm still looking for more explicit formulas or names of equations even... Incidentally, do you have a source for the $\mathbb{R}$-translations and how they give generalizations of the angle addition formulas? – James S. Cook Aug 2 '12 at 1:07
@James: nope. This is material that ought to be worked out immediately after the definition of the universal enveloping algebra in a textbook on Lie theory, but the textbooks on Lie theory that I've seen have other interests... – Qiaochu Yuan Aug 2 '12 at 1:25

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