nLab differential operator

Definition

We define general non-linear differential operators.

In differential geometry

Depending on which definition of differential operators one regards as fundamental, the following are either definitions or are propositions.

Definition/Proposition

For $X$ a smooth manifold and $(E\to X)$ a smooth bundle over $X$, write $(Jet(E)\to X)$ for its jet bundle.

For $(E_1 \to X)$, $(E_2 \to X)$ two bundles over $X$, then a differential operator

$$D \colon \Gamma_X(E_1) \to \Gamma_X(E_2)$$

between their spaces of sections is equivalently a map of the form

$$\phi \mapsto \tilde D \circ j_\infty(\phi)$$

where $j_\infty(\phi) \in \Gamma_X(Jet(E_1))$ is the jet prolongation of the section $\phi \in \Gamma_X(E_1)$, and where

$$\tilde D \colon Jet(E_1) \to E_2$$

is a bundle morphism from the jet bundle of $E_1$ to the bundle $E_2$.

In this form this appears for instance as (Saunders 89, def. 6.2.22). Discussion showing the equivalence of this definition with the maybe more traditional definition is in (Krasil’shchikVerbovetsky 98, def. 1.1, prop. 1.1, prop. 1.9, Krasilshchik 99, theorem 10).

Remark

The jet bundle construction $Jet \colon \mathbf{H}_{/X} \to \mathbf{H}_{/X}$ is (by the discussion there) a comonad on the category of bundles over $X$. In terms of this def. says that a differential operators from a bundle $E_1$ to a bundle $E_2$ is a morphism from $E_1$ to $E_2$ in the co-Kleisli category of the jet comonad.

Indeed, also the composition of differential operators is the composition in this co-Kleisli category (e.g. Marvan 93, section 1.1):

Proposition

The composition $D_2 \circ D_1 \colon \Gamma_X(E_1) \to \Gamma_X(E_3)$ of two differential operators $D_1 \colon \Gamma_X(E_1) \to \Gamma_X(E_2)$ and $D_2 \colon \Gamma_X(E_2)\to \Gamma_X(E_3)$ , def. , is given,under the identification of def. , by the composite

$$\widetilde{D_2 \circ D_1} \;\colon\; Jet(E_1) \longrightarrow Jet(Jet(E_1)) \stackrel{Jet(\tilde D_1)}{\longrightarrow} Jet(E_2) \stackrel{\tilde D_2}{\longrightarrow} E_3 \,,$$

where the first morphism is the counit of the jet bundle comonad.

Proof

Abbreviating $P_i \coloneqq \Gamma_X(E_i)$ and $J^\infty(P_i) = \Gamma_X(Jet(E_i))$, consider the following pasting diagram:

$$\array{ P_1 &\stackrel{id}{\longrightarrow}& P_1 &\stackrel{D_1}{\longrightarrow}& P_2 &\stackrel{D_2}{\longrightarrow}& P_3 \\ \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{j_\infty}} && \downarrow^{\mathrlap{j_\infty}} && \downarrow^{\mathrlap{id}} \\ P_1 &\stackrel{j_\infty}{\longrightarrow}& J^\infty(P_1) &\stackrel{J^\infty(D_1)}{\longrightarrow}& J^\infty(P_2) \\ \downarrow^{\mathrlap{j_\infty}} && \downarrow^{\mathrlap{j_\infty}} && \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{id}} \\ J^\infty (P_1) &\stackrel{c^{\infty,\infty}}{\longrightarrow}& J^\infty(J^\infty(P_1)) &\stackrel{J^\infty (\tilde D_1)}{\longrightarrow}& J^\infty(P_2) &\stackrel{\tilde D_2}{\longrightarrow}& P_3 } \,.$$

Here all the nontrivial squares are as in (Krasil’shchik-Verbovetsky 98, p. 12-13), with the bottom middle square being the image under $J^\infty$ of the square defining $\tilde D_1$. The bottom horizontal fillers of these squares are unique by (Krasil’shchik 99, theorem 10) (which is just our def/prop. ), hence the identification of the middle bottom morphism as displayed in the diagram.

With this, the morphism that our proposition claims is the correct composite is the total bottom morphism, and the differential operator that this defines by def. is the further composite with the left vertical morphism. Therefore the commutativity of the total diagram gives that this is equal to the total top morphisms, which is the composite of the two differential operators as claimed.

The co-Kleisli-like composition for finite order differential operators also appears in (Kock 10, section 7.3), from a perspective of synthetic differential geometry.

In differential cohesion

In view of the above one may axiomatize the category of differential operators in any context $\mathbf{H}$ of differential cohesion with infinitesimal shape modality $\Im$ as being the co-Kleisli category of the jet comonad

$$Jet_X \coloneqq i^\ast i_\ast$$

induced by base change along the unit $i \colon X \to \Im X$, for any choice of base space $X \in \mathbf{H}$.

In algebraic geometry / D-geometry

For the case of algebraic geometry, where $\Im X$ is known as the de Rham stack of a scheme $X$, and the quasicoherent sheaves on $i^\ast i_\ast X$ are the D-modules over $X$ (see at jet bundle for more on this), this statement is implicit in (Saito 89, def. 1.3).

Properties

Proposition

(differential operator preserves or shrinks wave front set)

Let $P$ be a differential operator (with smooth coefficients). Then for $u \in \mathcal{D}'$ a distribution, the wave front set of the derivative of distributions $P u$ is contained in the original wave front set of $u$:

$$WF(P u) \subset WF(u) \,.$$

(Hörmander 90, (8.1.11))

Types of differential operators

• elliptic differential operator

• regular differential operator, regular differential operator in noncommutative geometry

• pseudodifferential operator

• chiral differential operator

• crystalline differential operator

Related concepts

• formally adjoint differential operator

• symbol of a differential operator

• index of a differential operator?

• formal adjoint differential operator

• D-module

• Spencer cohomology

References

• Michal Marvan, A note on the category of partial differential equations, in Differential geometry and its applications, Proceedings of the Conference August 24-30, 1986, Brno (pdf)

  (notice that prop. 1.3 there holds only when the equalizer exists in the first place)

• David Saunders, The geometry of jet bundles, London Mathematical Society Lecture Note Series 142, Cambridge Univ. Press 1989.

• Morihiko Saito, Induced D-modules and differential complexes, Bull. Soc. Math. France 117 (1989), 361–387, pdf

• Lars Hörmander, The analysis of linear partial differential operators, vol. I, Springer 1983, 1990

• Michal Marvan, On Zero-Curvature Representations of Partial Differential Equations, (1993) (web)

• Joseph Krasil'shchik, Alexander Verbovetsky, Homological Methods in Equations of Mathematical Physics, Lectures given in August 1998 at the International Summer School in Levoca, Slovakia (arXiv:math/9808130)

• Joseph Krasil'shchik in collaboration with Barbara Prinari, Lectures on Linear Differential Operators over Commutative Algebras (pdf)

Discussion from a point of view of synthetic differential geometry is in

• Anders Kock, Synthetic geometry of manfiolds, Cambridge Tracts in Mathematics 180 (2010). (pdf)