# Principal symbol for non-linear differential operators

$\newcommand{\R}{\mathrm{R}} \newcommand{\N}{\mathrm{N}}\newcommand{\DD}{\mathrm{D}}\newcommand{\dd}{\mathrm{d}}$

Prerequisites: Let $\mathrm{T}: C^\infty(\Omega) \rightarrow C^\infty(\Omega), u(\cdot)\mapsto F(\cdot, \{\partial^\alpha u(\cdot)\}_{\vert{\alpha}\vert\leq k})$ be a non-linear differential operator, where $\Omega\subseteq\R^n$ is an open domain, $F:\R^N \rightarrow \R$ a smooth function and $\alpha \in \N_0^n$ denotes a multiindex.

Question: Is there a generally accepted way of defining the principal symbol of such a non-linear differential operator? If there is, how does it extend to systems of PDE's (i.e. $F$ is vector-valued)?

Approaches:

1. One (rather obvious) idea consists in linearizing $F$ in the $u$-dependent variables and then defining the $m$-th symbol (at $u$!) via $\sigma_m(T,u)(x,\xi) := \sigma_m(\DD_uT)(x,\xi)$ with the Frechet-derivative $\DD$.

2. Another is in using the Definition $\sigma_m(D)(x,\xi) := i^m \lim_{t\rightarrow\infty} \frac{1}{t^m} e^{-itf}\circ T\circ e^{itf}$ with $f\colon \R^n\rightarrow \R$ such that $\dd f(x)=\xi$. I am having problems with the well-definedness of this generalization. However, if this is a valid approach, am I correct in suspecting that this works out for every local operator?

Sources on the linear case: For the definition in the linear case I consulted the Wikipedia page and this lecture notes on linear analysis on manifolds.

## 2 Answers

I've seen only the first. It is indeed used mostly for identifying whether a nonlinear PDE is elliptic, hyperbolic, or parabolic. If so, one can use the respective linear theory, along with the appropriate implicit function theorem to prove existence theorems. Look up fully nonlinear elliptic PDEs for one well studied area.

To my knowledge, the principal symbol of a non-linear differential operator is not discussed very often. When I have seen it discussed, the definition basically coincided with your approach 1. For example, this is the definition that you can find in $\S 7$ of

Goldschmidt, H., Integrability criteria for systems of nonlinear partial differential equations, J. Differ. Geom. 1, 269-307 (1967). ZBL0159.14101.

Note that the above reference refers to the kernel of your symbol, interpreted as a linear map $\sigma(D_u\mathrm{T})(x,\xi) \colon \mathbb{R} \otimes S^kT^* \to \mathbb{R}$, as the principal symbol, but the two notions are equivalent up to invertible transformations on the operator $\mathrm{T}$. This definition happens to be the right one in the context of the study of formal properties of a PDE when localized to a point $x$ and the value $u$ of the dependent variable. Perhaps in some other context a different definition would be more appropriate.

• Can you state a reason why is not discussed usually? Is it not the standard tool of checking/defining ellipticity or hyperbolicity? – Peter Wildemann Sep 18 '17 at 15:57
• @ntor, you mean other than it's an empirical fact? I would simply guess that this notion has so far found limited utility. – Igor Khavkine Sep 18 '17 at 16:17
• Unfortunately I am not able to find the definition you are pointing towards in your reference. I am not very familiar with the language used there – Peter Wildemann Sep 20 '17 at 9:39
• @ntor, Goldschmidt's article is rather dense but mostly self-contained, so the notation is introduced earlier in the article. For a bit more discussion of this point of view on PDEs, see the earlier question MO76620. Once you can read the notation, the definition you are looking for is the second displayed equation of $\S7$ in Goldschmidt. Note, as I have already mentioned, the symbol (space) $g_k$ refers to the kernel of what you defined as the symbol (map) $\sigma$. – Igor Khavkine Sep 20 '17 at 10:34