Differential operator

In mathematics, a differential operator is a function that assigns a function and contains the derivative with respect to one or more variables. In particular, differential operators degrade the regularity of the function to which they are applied.

Probably the most important differential operator is the ordinary derivative, i.e. the mapping \textstyle {\frac {\mathrm {d} }{\mathrm {d} x}}(pronounced: "d to dx") which f^{\prime }assigns to a differentiable function fits derivative

{\frac {\mathrm {d} }{\mathrm {d} x}}\colon f\mapsto {\frac {\mathrm {d} }{\mathrm {d} x}}f={\frac {\mathrm {d} f}{\mathrm {d} x}}=f'

Differential operators can be linked together. By omitting the function on which they act, one obtains pure operator equations.

There are different definitions of a differential operator, all of which are special cases or generalisations of each other. Since the most general formulation is correspondingly difficult to understand, different definitions with different generalities are given here. For example, ordinary differential operators consist of the concatenation of whole derivatives, while partial derivatives also appear in partial differential operators.

Unless otherwise stated, in this article let M\subset \mathbb {R} ^{n}a bounded and open set. Moreover, C^{k}(M)the set of k-times continuously differentiable functions f\colon M\to \mathbb {R} and C(M)=C^{0}(M)denotes the set of continuous functions. The restriction that fmaps between real subsets is not necessary, but is usually assumed in this article. If other definition and image ranges are necessary or useful, this is explicitly stated in the following.

This article is also largely restricted to differential operators operating on the spaces of continuously differentiable functions just mentioned. There are weakenings of the definitions. For example, the study of differential operators led to the definition of the weak derivative and thus to Sobolev spaces, which are a generalisation of the spaces of continuously differentiable functions. This led further to the idea of studying linear differential operators with the help of functional analysis in operator theory. However, these aspects will not be discussed further in this article for the time being. A generalisation of a differential operator is the pseudo-differential operator.

First order linear differential operator

Definition

Let M\subset \mathbb {R} ^{n}an open subset. A first order linear differential operator is a mapping

{\displaystyle D\colon C^{1}(M)\to C^{0}(M),}

which through

u\mapsto \sum _{i=1}^{n}a_{i}(x)\partial _{x_{i}}u

where a_{i}is a continuous function.

Examples

  • The most important example of a first order differential operator is the ordinary derivative

\frac{\mathrm{d}}{\mathrm{d} x}\colon f \mapsto f'.

  • The partial derivative

\frac{\partial}{\partial x_i}\colon f \mapsto \frac{\partial f}{\partial x_i}

in x_{i}-direction is a partial differential operator of first order.

  • Other differential operators of this kind are obtained by multiplication with a continuous function. If {\displaystyle a\in C^{0}(M)}just such a continuous function, then the operator given by

D = a \frac{\mathrm{d}}{\mathrm{d} x}\colon f \mapsto a f' \quad \text{d. h.} \quad Df(x) = a(x)f'(x)

operator Dis again a first-order differential operator.

  • Three further examples are the operators gradient (grad), divergence (div) and rotation (rot) from vector analysis. They are \nabla denoted by the Nabla symbol , which in the three-dimensional case in Cartesian coordinates has the form

\nabla = \begin{pmatrix}\frac{\partial}{\partial x_1} \\ \frac{\partial}{\partial x_2} \\ \frac{\partial}{\partial x_3}\end{pmatrix}.

has.

  • The Wirtinger derivations

{\displaystyle {\frac {\partial }{\partial z}}={\frac {1}{2}}\left({\frac {\partial }{\partial x}}-\mathrm {i} {\frac {\partial }{\partial y}}\right)}

and

{\displaystyle {\frac {\partial }{\partial {\overline {z}}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x}}+\mathrm {i} {\frac {\partial }{\partial y}}\right)}

are two more examples of differential operators. The special thing in these operators is that they can be used to calculate functions {\displaystyle M\subset \mathbb {C} \to \mathbb {C} }for holomorphism, namely if ∂ \textstyle \frac{\partial f}{\partial \overline{z}} = 0then the function is fholomorphic.

Ordinary differential operator

Ordinary differential operators occur in particular in connection with ordinary differential equations.

Definition

Analogous to the definition of the first-order differential operator, an ordinary differential operator of order kis a mapping

{\displaystyle D\colon C^{k}(M)\to C^{0}(M),}

which through

D(f)(x) := \sum_{i=0}^k a_i(x) \left(\frac{\mathrm{d}^i f}{\mathrm{d} x^i}(x)\right)^{\beta_i}

is given. Here a_{i}for all is iagain a continuous function. In the case β \beta_i = 1for all this operator is icalled an ordinary linear differential operator.

Example

  • The derivative k-th order

\frac{\mathrm{d}^k}{\mathrm{d} x^k}\colon f \mapsto f^{(k)}

is the simplest case of an ordinary differential operator. It is the special case \beta_k = 1from {\displaystyle a_{i}\equiv 0}for {\displaystyle i<k,\;a_{k}\equiv 1}k =

Linear partial differential operator

Definition

Let M\subset \mathbb {R} ^{n}an open subset. A linear partial differential operator of order kis a linear operator

{\displaystyle D\colon C^{k}(M)\to C^{0}(M),}

who through

D(f)(x) := \sum_{|\alpha| \leq k} a_{\alpha}(x) \frac{\partial^\alpha f}{\partial x^\alpha}(x)

can be represented. Where a_{\alpha}for all multi-indices α\alpha \in \N^nis a continuous function.

Examples

  • The Laplace operator in Cartesian coordinates is

\Delta= \nabla^2 = \sum_{k=1}^n \frac{\partial^2}{\partial x_k^2}.

This is an elementary example of a partial differential operator. Moreover, this is the most important example of an elliptic differential operator. Elliptic differential operators are a special class of partial differential operators.

  • The operator corresponding to the heat conduction or diffusion equation is

\Delta - \frac{\partial}{\partial t}.

This is an example of a parabolic differential operator.

  • The D'Alembert operator

\Box\varphi(x,y,z, t)=\frac{1}{c^2}\frac{\partial^2\varphi}{\partial t^2}(x,y,z,t) - \Delta_{(x,y,z)}\varphi(x,y,z,t),

where ccorresponds to a velocity, is another important partial differential operator. This is a hyperbolic operator and is used in the wave equation.

Partial differential operator

Definition

A (non-linear) partial differential operator of order kis again a mapping

{\displaystyle D\colon C^{k}(M)\to C^{0}(M).}

This is given by

D(f)(x) := \sum_{i} \sum_{|\alpha| \leq k} a_{\alpha i}(x) \left(\frac{\partial^\alpha f}{\partial x^\alpha}(x)\right)^{i}.

Here a_{\alpha i}for all α\alpha \in \N^nand icontinuous functions.

Linear differential operators

In the definitions above, it was already briefly mentioned when an ordinary or a partial differential operator is called linear. For the sake of completeness, the abstract definition of a linear differential operator is now given. This is analogous to the definition of the linear mapping. All the examples given above are linear differential operators, unless otherwise stated.

Definition

Let Dbe an (arbitrary) differential operator. This is called linear if

{D}\,(f+g) = ({D}f) + ({D}g)

{D}\,(cf) = c\,({D}f)

cholds for all functions f, g \in C^1(M)and all constants

The most prominent example of this is the differential operator

\frac{\mathrm{d}}{\mathrm{d} x}\colon f \mapsto f',

which fassigns its derivative to a function .

The solution space of a linear differential equation forms a vector space. After Fourier transformation, they can often be traced back to algebraic equations and concepts of linear algebra. Non-linear differential operators are much more difficult to deal with.

Algebra of the differential operators

\operatorname{Diff}^k(C^k(M))kdenotes the set of all linear differential operators of order C^{k}(M)operating on . The set

\operatorname{Diff}(C^k(M)) := \bigoplus_{k \geq 0} \operatorname{Diff}^k(C^k(M))

together with the series connection of linear differential operators is called multiplication

(\mathrm{D}_1\circ \mathrm{D}_2)(f) = \mathrm{D}_1(\mathrm{D}_2(f))

to a \Z_+-graded algebra. However, multiplication is generally not commutative. An exception are, for example, differential operators with constant coefficients, where commutativity follows from the interchangeability of the partial derivatives.

One can also formally Dbuild power series with the differential operators and exponential functions \exp (D)above them, for example. For calculating with such exponential expressions of linear operators, the Baker-Campbell-Hausdorff formulae apply.

Differential operator on a manifold

Since on manifolds one only has the local coordinate systems in the form of maps and no globally valid coordinate systems available, one must define coordinate-independent differential operators on these. Such differential operators on manifolds are also called geometric differential operators.

Coordinates-invariant definition

Let Mbe a smooth manifold and let E, F \to Mvector bundles. A differential operator of order kbetween the intersections of Eand Fis a linear mapping

D \colon \Gamma^\infty(M,E) \to \Gamma^{\infty}(M,F)

with the following characteristics:

  • The operator Dis local, i.e. the following applies

\operatorname{supp}(Ds) \subseteq \operatorname{supp}(s).

  • For x\in Mthere exist an open environment U\subseteq Mof x, bundle maps ϕ {\displaystyle \phi \colon E|_{U}\to U\times \mathbb {C} ^{r}}and ψ {\displaystyle \psi \colon F|_{U}\to U\times \mathbb {C} ^{s}}and a differential operator {\displaystyle {\tilde {D}}\in \operatorname {Diff} ^{k}(U,\mathbb {C} ^{r},\mathbb {C} ^{s}),}so that the diagram is
     {\displaystyle {\begin{array}{ccc}\Gamma _{0}^{\infty }(E\vert _{U})&{\xrightarrow {D}}&\Gamma _{0}^{\infty }(F\vert _{U})\\{\big \downarrow }\phi ^{*}&&{\big \downarrow }\psi ^{*}\\C^{\infty }(U,\mathbb {C} ^{r})&{\xrightarrow {\tilde {D}}}&C^{\infty }(U,\mathbb {C} ^{s})\end{array}}}
    . By ϕ    \phi^*is {\displaystyle C^{\infty }(U,\mathbb {C} ^{r})}denoted the pullback of a smooth vector field into the space

Examples

In the following, examples of geometric differential operators are shown.

  • The set of differential forms forms a smooth vector bundle over a smooth manifold. The Cartan derivative and its adjoint operator are differential operators on this vector bundle.
  • The Laplace-Beltrami operator and other generalised Laplace operators are differential operators.
  • The tensor bundle is a vector bundle. For any fixed chosen vector field Xthe mapping \nabla_X \colon \Gamma^\infty(T^k_lM) \rightarrow \Gamma^\infty(T^k_lM)defined by T \mapsto \nabla_X Twhere \nabla is the covariant derivative, a differential operator.
  • The Lie derivative is a differential operator on the differential forms.

Symbol of a differential operator

The 2nd order differential operators given in the examples, if one \partial _{i}formally y_{i}replaces the partial derivatives ∂ by variables and considers only the highest - i.e. second - order terms, correspond to a quadratic form in the y_{i}. In the elliptic case all coefficients of the form have the same sign, in the hyperbolic case the sign changes, in the parabolic case the highest order term is missing for one of the y_{i}The corresponding partial differential equations each show very different behaviour. The names come from the analogues to conic section equations.

This can be extended to other cases by the notion of the main symbol of the differential operator. One keeps only terms of the highest order, replaces derivatives by new variables y_{i}and obtains a polynomial in these new variables with which one can characterise the differential operator. For example, it is of the elliptic type if the following holds: the principal symbol is non-zero if at least one y_{i}non-zero. However, there are already "mixed" cases with 2nd order differential operators that cannot be assigned to any of the three classes.

The following definitions capture this again in mathematical precision.

Symbol

Let it be

P(u)(x) = \sum_{|\alpha|\leq m} b_\alpha(x) \frac{\partial^\alpha}{\partial x^\alpha}u(x)

a general differential operator of order m. The coefficient function b_{\alpha} \in C^\infty(\R^n)can be matrix-valued. The polynomial

p(x,\xi) = \sum_{|\alpha|\leq m}b_\alpha(x) \left(i\xi\right)^\alpha

in ξ\xi \in \R^nis called the symbol of P. However, since, as already indicated in the introduction, the most important information is to be found in the highest order term, the following definition of the main symbol is usually used.

Main icon

Let Pagain be the differential operator of order mdefined above. The homogeneous polynomial

p_m(x,\xi) = \sum_{|\alpha|=m}b_\alpha(x) \left(i\xi\right)^\alpha

in ξ\xi \in \R^nis called the main symbol of P. Often the main symbol is simply called symbol if confusion with the definition given above is excluded.

Examples

  • The symbol and the main symbol of the Laplace operator Δ \Delta are as follows

\sum_{i=1}^n -\xi_i^2 = -|\xi|^2.

Main symbol of a differential operator between vector bundles

Differential operators on manifolds can also be assigned a symbol and a main symbol. Of course, it must be taken into account in the definition that the main symbol and the symbol under map change are defined invariant. Since the map change for symbols is very complicated, one usually restricts oneself to the definition of the main symbol.

Let D \colon \Gamma^\infty(M,E) \to \Gamma^\infty(M,F)be a (coordinate-invariant) differential operator operating between intersections of vector bundles. Let p\in M, ξ\xi \in T_p^*Mand e \in E_p. Choose f \in C^\infty_c(M)and s \in \Gamma^\infty_c(M,E)with f(p) = 0, \textstyle\mathrm{d}f_p = \xiand s(p) = e. Then the expression

\sigma^k_D(p,\xi) e := \frac{i^k}{k!}D(f^k s)(p)

independent of the choice of fand s. The function

\sigma_D^k(p, \xi) \in \operatorname{Hom}(E_p,F_p)

is then called the main symbol of D.

Pseudo-differential operators

→ Main article: Pseudo-differential operator

The order of a differential operator is always integer and positive. In the theory of pseudo-differential operators this is generalised. Linear differential operators of order kwith smooth and bounded coefficients can be understood as pseudo-differential operators of the same order. Let D \colon C^k_c(\R^n) \to C_c(\R^n)such a differential operator, then one can \mathcal{F}^{-1}apply to Dfthe Fourier transform {\mathcal {F}}and then the inverse Fourier transform That is to say

{\displaystyle (Du)(x)=({\mathcal {F}}^{-1}{\mathcal {F}}Du)(x)={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}\int _{\mathbb {R} ^{n}}e^{\mathrm {i} (x-y)\xi }D(\xi )u(y)\mathrm {d} y\mathrm {d} \xi .}

This is a special case of a pseudo-differential operator

{\displaystyle (Pu)(x)={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}\int _{\mathbb {R} ^{n}}e^{\mathrm {i} (x-y)\xi }a(x,y,\xi )u(y)\mathrm {d} y\mathrm {d} \xi .}

This also shows that certain differential operators can be represented as integral operators and thus differential operators and integral operators are not completely opposed.


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