Differential operator

In mathematics, a differential operator is a function that assigns a function to a function and contains the derivative according 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; H. the mapping (spoken: "d to dx"), which assigns its derivative to a differentiable function : ${\displaystyle \textstyle {\frac {\mathrm {d} }{\mathrm {d} x}}}$ ${\displaystyle f}$ ${\displaystyle f^{\prime }}$

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

Differential operators can be linked with one another. By omitting the function on which they act, pure operator equations are obtained.

There are different definitions of a differential operator, all of which are special cases or generalizations of one another. Since the most general formulation is accordingly difficult to understand, different definitions with different general validity are given here. Ordinary differential operators consist of the concatenation of whole derivatives, while partial derivatives also appear in partial differential operators.

Unless otherwise stated, this article is a restricted and open set . In addition, with denotes the set of- times continuously differentiable functions and with the set of continuous functions. The restriction that mapping between real subsets is not necessary, but is usually assumed in this article. If other definition and image areas are necessary or useful, this is explicitly stated below. ${\displaystyle M\subset \mathbb {R} ^{n}}$ ${\displaystyle C^{k}(M)}$ ${\displaystyle k}$ ${\displaystyle f\colon M\to \mathbb {R} }$ ${\displaystyle C(M)=C^{0}(M)}$ ${\displaystyle f}$

This article is also largely restricted to differential operators that operate on the spaces of continuously differentiable functions just mentioned. There are weakening of the definitions. For example, the study of differential operators led to the definition of the weak derivative and thus to the Sobolev spaces , which are a generalization of the spaces of continuously differentiable functions. This led to the idea of investigating 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 generalization of a differential operator is the pseudo-differential operator .

First order linear differential operator

definition

Be an open subset . A first order linear differential operator is a map ${\displaystyle M\subset \mathbb {R} ^{n}}$

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

by

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

can be represented, where is a continuous function . ${\displaystyle a_{i}}$

Examples

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

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

• The partial derivative

${\displaystyle {\frac {\partial }{\partial x_{i}}}\colon f\mapsto {\frac {\partial f}{\partial x_{i}}}}$

in -direction is a first order partial differential operator. ${\displaystyle x_{i}}$

• Other differential operators of this kind can be obtained by multiplying by a continuous function. Let it be a continuous function, then it is through ${\displaystyle a\in C^{0}(M)}$

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

defined operator again a first order differential operator. ${\displaystyle D}$

• Three further examples are the operators gradient (grad), divergence (div) and rotation (red) from vector analysis . They are denoted by the Nabla symbol , which in the three-dimensional case represents the shape in Cartesian coordinates ${\displaystyle \nabla }$

${\displaystyle \nabla ={\begin{pmatrix}{\frac {\partial }{\partial x_{1}}}\\{\frac {\partial }{\partial x_{2}}}\\{\frac {\partial }{\partial x_{3}}}\end{pmatrix}}.}$

Has.

• The Wirtinger derivatives

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

and

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

are two other examples of differential operators. The special thing about these operators is that you can use them to examine functions for holomorphism , because then the function is holomorphic. ${\displaystyle M\subset \mathbb {C} \to \mathbb {C} }$ ${\displaystyle \textstyle {\frac {\partial f}{\partial {\overline {z}}}}=0}$ ${\displaystyle f}$

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 order differential operator is a mapping ${\displaystyle k}$

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

by

${\displaystyle D(f)(x):=\sum _{i=0}^{k}a_{i}(x)\left({\frac {\mathrm {d} ^{i}f}{\mathrm {d} x^{i}}}(x)\right)^{\beta _{i}}}$

given is. Here is another continuous function for everyone . In the case for all , this operator is called an ordinary, linear differential operator. ${\displaystyle a_{i}}$ ${\displaystyle i}$ ${\displaystyle \beta _{i}=1}$ ${\displaystyle i}$

example

• The -th order derivative ${\displaystyle k}$

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

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

Linear partial differential operator

definition

Be an open subset. A linear partial differential operator of order is a linear operator ${\displaystyle M\subset \mathbb {R} ^{n}}$ ${\displaystyle k}$

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

the through

${\displaystyle D(f)(x):=\sum _{|\alpha |\leq k}a_{\alpha }(x){\frac {\partial ^{\alpha }f}{\partial x^{\alpha }}}(x)}$

can be represented. Where is a continuous function for all multi-indices . ${\displaystyle a_{\alpha }}$ ${\displaystyle \alpha \in \mathbb {N} ^{n}}$

Examples

• The Laplace operator in Cartesian coordinates is

${\displaystyle \Delta =\nabla ^{2}=\sum _{k=1}^{n}{\frac {\partial ^{2}}{\partial x_{k}^{2}}}.}$

This is an elementary example of a partial differential operator. It is also 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

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

This is an example of a parabolic differential operator.

• The D'Alembert operator

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

where corresponds to speed is another important partial differential operator. This is a hyperbolic operator and is used in the wave equation . ${\displaystyle c}$

Partial differential operator

definition

A (non-linear) partial differential operator of order is also a mapping again ${\displaystyle k}$

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

This is given by

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

Here are for everyone and continuous functions. ${\displaystyle a_{\alpha i}}$ ${\displaystyle \alpha \in \mathbb {N} ^{n}}$ ${\displaystyle i}$

Linear differential operators

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

definition

Let be an (arbitrary) differential operator. This is called linear if ${\displaystyle D}$

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

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

holds for all functions and all constants . ${\displaystyle f,g\in C^{1}(M)}$ ${\displaystyle c}$

The most prominent example of this is the differential operator

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

which assigns its derivative to a function . ${\displaystyle f}$

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. Nonlinear differential operators are much more difficult to deal with.

Algebra of the differential operators

The set of all linear differential operators of the order that operate on is denoted by . The amount ${\displaystyle \operatorname {Diff} ^{k}(C^{k}(M))}$ ${\displaystyle k}$ ${\displaystyle C^{k}(M)}$

${\displaystyle \operatorname {Diff} (C^{k}(M)):=\bigoplus _{k\geq 0}\operatorname {Diff} ^{k}(C^{k}(M))}$

is used together with the series connection of linear differential operators as multiplication

${\displaystyle (\mathrm {D} _{1}\circ \mathrm {D} _{2})(f)=\mathrm {D} _{1}(\mathrm {D} _{2}(f))}$

to one - graduate algebra . In general, however, multiplication is not commutative. An exception are, for example, differential operators with constant coefficients, in which the commutativity follows from the interchangeability of the partial derivatives . ${\displaystyle \mathbb {Z} _{+}}$

One can also formally form power series with the differential operators and use z. B. Exponential Functions . The Baker-Campbell-Hausdorff formulas apply to computing with such exponential expressions of linear operators . ${\displaystyle D}$ ${\displaystyle \exp(D)}$

Differential operator on a manifold

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

Coordinate-invariant definition

Let be a smooth manifold and be vector bundles . A differential operator of the order between the sections of and is a linear mapping ${\displaystyle M}$ ${\displaystyle E,F\to M}$ ${\displaystyle k}$ ${\displaystyle E}$ ${\displaystyle F}$

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

with the following characteristics:

• The operator is local, that is, it holds ${\displaystyle D}$

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

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

Examples

Examples of geometric differential operators are shown below.

• 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, as well as other generalized Laplace operators, are differential operators.
• The tensor bundle is a vector bundle. For each fixed vector field , the mapping is defined by , where the covariant derivative is, a differential operator. ${\displaystyle X}$ ${\displaystyle \nabla _{X}\colon \Gamma ^{\infty }(T_{l}^{k}M)\rightarrow \Gamma ^{\infty }(T_{l}^{k}M)}$ ${\displaystyle T\mapsto \nabla _{X}T}$ ${\displaystyle \nabla }$
• The Lie derivative is a differential operator on the differential forms.

Symbol of a differential operator

If the partial derivatives are formally replaced by variables and only the terms of the highest - i.e. second - order are considered, the second order differential operators given in the examples correspond to a quadratic form in the . In the elliptical 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 coefficients . The corresponding partial differential equations each show very different behavior. The names come from the analogues of conic section equations . ${\displaystyle \partial _{i}}$ ${\displaystyle y_{i}}$ ${\displaystyle y_{i}}$ ${\displaystyle y_{i}}$

This can also be extended to other cases using the concept of the main symbol of the differential operator. Only terms of the highest order are retained, derivatives are replaced by new variables, and a polynomial is obtained in these new variables with which the differential operator can be characterized. For example, it is of the elliptical type if: the main symbol is not equal to zero, if at least one is not equal to zero. However, there are already “mixed” cases for differential operators of the 2nd order that cannot be assigned to any of the three classes. ${\displaystyle y_{i}}$ ${\displaystyle y_{i}}$

The following definitions record this again with mathematical precision.

symbol

Be it

${\displaystyle P(u)(x)=\sum _{|\alpha |\leq m}b_{\alpha }(x){\frac {\partial ^{\alpha }}{\partial x^{\alpha }}}u(x)}$

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

${\displaystyle p(x,\xi )=\sum _{|\alpha |\leq m}b_{\alpha }(x)\left(i\xi \right)^{\alpha }}$

in is the symbol of . However, since the most important information can be found in the term of the highest order, as already indicated in the introduction, the following definition of the main symbol is usually used. ${\displaystyle \xi \in \mathbb {R} ^{n}}$ ${\displaystyle P}$

Main symbol

Let again be the differential operator of order defined above . The homogeneous polynomial ${\displaystyle P}$ ${\displaystyle m}$

${\displaystyle p_{m}(x,\xi )=\sum _{|\alpha |=m}b_{\alpha }(x)\left(i\xi \right)^{\alpha }}$

in is the main symbol of . Often the main symbol is simply called a symbol if confusion with the definition given above is impossible. ${\displaystyle \xi \in \mathbb {R} ^{n}}$ ${\displaystyle P}$

Examples

• The symbol and the main symbol of the Laplace operator are ${\displaystyle \Delta }$

${\displaystyle \sum _{i=1}^{n}-\xi _{i}^{2}=-|\xi |^{2}.}$

Main symbol of a differential operator between vector bundles

Differential operators on manifolds can also be assigned a symbol and a main symbol. In the definition, it must of course be taken into account that the main symbol and the symbol under Change of cards are invariant defined. Since changing cards for symbols is very complicated, you usually limit yourself to the definition of the main symbol.

Let be a (coordinate-invariant) differential operator that operates between intersections of vector bundles. Be , and . Choose and with , and . Then the expression ${\displaystyle D\colon \Gamma ^{\infty }(M,E)\to \Gamma ^{\infty }(M,F)}$ ${\displaystyle p\in M}$ ${\displaystyle \xi \in T_{p}^{*}M}$ ${\displaystyle e\in E_{p}}$ ${\displaystyle f\in C_{c}^{\infty }(M)}$ ${\displaystyle s\in \Gamma _{c}^{\infty }(M,E)}$ ${\displaystyle f(p)=0}$ ${\displaystyle \textstyle \mathrm {d} f_{p}=\xi }$ ${\displaystyle s(p)=e}$

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

regardless of the choice of and . The function ${\displaystyle f}$ ${\displaystyle s}$

${\displaystyle \sigma _{D}^{k}(p,\xi )\in \operatorname {Hom} (E_{p},F_{p})}$

is then called the main symbol of . ${\displaystyle D}$

Pseudo differential operators

→ Main article : Pseudo-differential operator

The order of a differential operator is always an integer and positive. This is generalized in the theory of pseudo-differential operators. Linear differential operators of the order with smooth and bounded coefficients can be understood as pseudo-differential operators of the same order. If such a differential operator is used, then one can apply the Fourier transformation and then the inverse Fourier transformation . That is, it applies ${\displaystyle k}$ ${\displaystyle D\colon C_{c}^{k}(\mathbb {R} ^{n})\to C_{c}(\mathbb {R} ^{n})}$ ${\displaystyle Df}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle {\mathcal {F}}^{-1}}$

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

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 opposite.

literature

• Otto Forster : Analysis 2. Differential calculus in R ⁿ . Ordinary differential equations. Vieweg-Verlag, 7th edition, 2006, ISBN 3-528-47231-6 .
• Konrad Königsberger : Analysis 2. Springer-Verlag, Berlin / Heidelberg 2000, ISBN 3-540-43580-8 .
• Dirk Werner : Functional Analysis. Springer-Verlag, Berlin 2007, ISBN 978-3-540-72533-6 .
• Lawrence Evans : Partial Differential Equations. American Mathematical Society, ISBN 0-8218-0772-2 .
• Liviu I. Nicolaescu: Lectures on the geometry of manifolds. World Scientific Pub Co (for differential operators between vector bundles).