# Green function

A function related to integral representations of solutions of boundary value problems for differential equations.

The Green function of a boundary value problem for a linear differential equation is the fundamental solution of this equation satisfying homogeneous boundary conditions. The Green function is the kernel of the integral operator inverse to the differential operator generated by the given differential equation and the homogeneous boundary conditions (cf. Kernel of an integral operator). The Green function yields solutions of the inhomogeneous equation satisfying the homogeneous boundary conditions. Finding the Green function reduces the study of the properties of the differential operator to the study of similar properties of the corresponding integral operator.

## Green function for ordinary differential equations.

Let $L$ be the differential operator generated by the differential polynomial

$$l [ y] = \ \sum _ {k = 0 } ^ { n } p _ {k} ( x) \frac{d ^ {k} y }{dx ^ {k} } ,\ \ a < x < b,$$

and the boundary conditions $U _ {j} [ y] = 0$, $j = 1 \dots n$, where

$$U _ {j} [ y] = \ \sum _ {k = 0 } ^ { n } \alpha _ {jk} y ^ {(} k) ( a) + \beta _ {jk} y ^ {(} k) ( b).$$

The Green function of $L$ is the function $G ( x, \xi )$ that satisfies the following conditions:

1) $G ( x, \xi )$ is continuous and has continuous derivatives with respect to $x$ up to order $n - 2$ for all values of $x$ and $\xi$ in the interval $[ a, b]$.

2) For any given $\xi$ in $( a, b)$ the function $G ( x, \xi )$ has uniformly-continuous derivatives of order $n$ with respect to $x$ in each of the half-intervals $[ a, \xi )$ and $( \xi , b]$ and the derivative of order $n - 1$ satisfies the condition

$$\frac{\partial ^ {n - 1 } }{\partial x ^ {n - 1 } } G ( \xi + , \xi ) - \frac{\partial ^ {n - 1 } }{\partial x ^ {n - 1 } } G ( \xi - , \xi ) = \ \frac{1}{p _ {n} ( \xi ) }$$

if $x = \xi$.

3) In each of the half-intervals $[ a, \xi )$ and $( \xi , b]$ the function $G ( x, \xi )$, regarded as a function of $x$, satisfies the equation $l [ G] = 0$ and the boundary conditions $U _ {j} [ G] = 0$, $j = 1 \dots n$.

If the boundary value problem $Ly = 0$ has trivial solutions only, then $L$ has one and only one Green function . For any continuous function $f$ on $[ a, b]$ there exists a solution of the boundary value problem $Ly = f$, and it can be expressed by the formula

$$y ( x) = \ \int\limits _ { a } ^ { b } G ( x, \xi ) f ( \xi ) d \xi .$$

If the operator $L$ has a Green function $G ( x, \xi )$, then the adjoint operator $L ^ {*}$ also has a Green function, equal to $\overline{ {G ( \xi , x) }}\;$. In particular, if $L$ is self-adjoint ( $L = L ^ {*}$), then $G ( x, \xi ) = \overline{ {G ( \xi , x) }}\;$, i.e. the Green function is a Hermitian kernel in this case. Thus, the Green function of the self-adjoint second-order operator $L$ generated by the differential operator with real coefficients

$$l [ y] = \ \frac{d}{dx} \left ( p \frac{dy }{dx } \right ) + q ( x) y,\ \ a < x < b,$$

and the boundary conditions $y ( a) = 0$, $y ( b) = 0$ has the form:

$$G ( x, \xi ) = \ \left \{ \begin{array}{ll} Cy _ {1} ( x) y _ {2} ( \xi ) &\textrm{ if } x \leq \xi , \\ Cy _ {1} ( \xi ) y _ {2} ( x) &\textrm{ if } x > \xi . \\ \end{array} \right .$$

Here $y _ {1} ( x)$ and $y _ {2} ( x)$ are arbitrary independent solutions of the equation $l [ y] = 0$ satisfying, respectively, the conditions $y _ {1} ( a) = 0$, $y _ {2} ( b) = 0$; $C = [ p ( \xi ) W ( \xi )] ^ {-} 1$, where $W$ is the Wronski determinant (Wronskian) of $y _ {1}$ and $y _ {2}$. It can be shown that $C$ is independent of $\xi$.

If the operator $L$ has a Green function, then the boundary eigen value problem $Ly = \lambda y$ is equivalent to the integral equation $y ( x) = \lambda \int _ {a} ^ {b} G ( x, \xi ) y ( \xi ) d \xi$, to which Fredholm's theory is applicable (cf. also Fredholm theorems). For this reason the boundary value problem $Ly = \lambda y$ can have at most a countable number of eigen values $\lambda _ {1} , \lambda _ {2} \dots$ without finite limit points. The conjugate problem has complex-conjugate eigen values of the same multiplicity. For each $\lambda$ that is not an eigen value of $L$ it is possible to construct the Green function $G ( x, \xi , \lambda )$ of the operator $L - \lambda I$, where $I$ is the identity operator. The function $G ( x, \xi , \lambda )$ is a meromorphic function of the parameter $\lambda$; its poles can be eigen values of $L$ only. If the multiplicity of the eigen value $\lambda _ {0}$ is one, then

$$G ( x, \xi , \lambda ) = \ \frac{u _ {0} ( x) \overline{ {v _ {0} ( \xi ) }}\; }{\lambda - \lambda _ {0} } + G _ {1} ( x, \xi , \lambda ),$$

where $G _ {1} ( x, \xi , \lambda )$ is regular in a neighbourhood of the point $\lambda _ {0}$, and $u _ {0} ( x)$ and $v _ {0} ( x)$ are the eigen functions of $L$ and $L ^ {*}$ corresponding to the eigen values $\lambda _ {0}$ and $\overline{ {\lambda _ {0} }}\;$ and normalized so that

$$\int\limits _ { a } ^ { b } u _ {0} ( x) \overline{ {v _ {0} ( x) }}\; dx = 1.$$

If $G ( x, \xi , \lambda )$ has infinitely-many poles and if these are of the first order only, then there exists a complete biorthogonal system

$$u _ {1} ( x),\ u _ {2} ( x) ,\dots ; \ \ v _ {1} ( x),\ v _ {2} ( x) \dots$$

of eigen functions of $L$ and $L ^ {*}$. If the eigen values are numbered in increasing sequence of their absolute values, then the integral

$$I _ {R} ( x, f ) = \ \frac{1}{2 \pi i } \int\limits _ {| \lambda | = R } \ d \lambda \int\limits _ { a } ^ { b } G ( x, \xi , \lambda ) f ( \xi ) d \xi$$

is equal to the partial sum

$$S _ {k} ( x, f ) = \ \sum _ {| \lambda _ {n} | < R } u _ {n} ( x) \int\limits _ { a } ^ { b } f ( \xi ) \overline{ {v _ {n} ( \xi ) }}\; \ d \xi$$

of the expansion of $f$ with respect to the eigen functions of $L$. The positive number $R$ is so selected that the function $G ( x, \xi , \lambda )$ is regular in $\lambda$ on the circle $| \lambda | = R$. For a regular boundary value problem and for any piecewise-smooth function $f$ in the interval $a < x < b$, the equation

$$\lim\limits _ {R \rightarrow \infty } \ I _ {R} ( x, f ) = \ { \frac{1}{2} } [ f ( x + 0) + f ( x - 0)]$$

is valid, that is, an expansion into a convergent series is possible .

If the Green function $G ( x, \xi , \lambda )$ of the operator $L - \lambda I$ has multiple poles, then its principal part is expressed by canonical systems of eigen and adjoint functions of the operators $L$ and $L ^ {*}$.

In the case considered above, the boundary value problem $Ly = 0$ has no non-trivial solutions. If, on the other hand, such non-trivial solutions exist, a so-called generalized Green function is introduced. Let there exist, e.g., exactly $m$ linearly independent solutions of the problem $Ly = 0$. Then a generalized Green function $\widetilde{G} ( x, \xi )$ exists that has properties 1) and 2) of an ordinary Green function, satisfies the boundary conditions as a function of $x$ if $a < \xi < b$ and, in addition, is a solution of the equation

$$l _ {x} [ y] = - \sum _ {k = 1 } ^ { m } \phi _ {k} ( x) \overline{ {v _ {k} ( \xi ) }}\; .$$

Here $\{ v _ {k} ( x) \} _ {k = 1 } ^ {m}$ is a system of linearly independent solutions of the adjoint problem $L ^ {*} y = 0$, while $\{ \phi _ {k} ( x) \} _ {k = 1 } ^ {m}$ is an arbitrary system of continuous functions biorthogonal to it. Then

$$y ( x) = \ \int\limits _ { a } ^ { b } \widetilde{G} ( x, \xi ) f ( \xi ) d \xi$$

is the solution of the boundary value problem $Ly = f$ if the function $f$ is continuous and satisfies the solvability criterion, i.e. is orthogonal to all $v _ {k}$.

If $\widetilde{G} _ {0}$ is one of the generalized Green functions of $L$, then any other generalized Green function can be represented in the form

$$\widetilde{G} ( x, \xi ) = \ \widetilde{G} _ {0} ( x, \xi ) + \sum _ {k = 1 } ^ { m } u _ {k} ( x) \psi _ {k} ( \xi ),$$

where $\{ u _ {k} ( x) \}$ is a complete system of linearly independent solutions of the problem $Ly = 0$, and $\psi _ {k} ( \xi )$ are arbitrary continuous functions .

## Green function for partial differential equations.

1) Elliptic equations. Let $A$ be the elliptic differential operator of order $m$ generated by the differential polynomial

$$a ( x, D) = \ \sum _ {| \alpha | \leq m } a _ \alpha ( x) D ^ \alpha$$

in a bounded domain $\Omega \subset \mathbf R ^ {N}$ and the homogeneous boundary conditions $B _ {j} u = 0$, where $B _ {j}$ are boundary operators with coefficients defined on the boundary $\partial \Omega$ of $\Omega$, which is assumed to be sufficiently smooth. A function $G ( x, y)$ is said to be a Green function for $A$ if, for any fixed $y \in \Omega$, it satisfies the homogeneous boundary conditions $B _ {j} G ( x, y) = 0$ and if, regarded as a generalized function, it satisfies the equation

$$a ( x, D) G ( x, y) = \ \delta ( x - y).$$

In the case of operators with smooth coefficients and normal boundary conditions, which ensure that the solution of the homogeneous boundary value problem is unique, a Green function exists and the solution of the boundary value problem $Au = f$ can be represented in the form (cf. )

$$u ( x) = \ \int\limits _ \Omega G ( x, y) f ( y) dy.$$

In such a case the uniform estimates for $x , y \in \overline \Omega \;$,

$$| G ( x, y) | \leq \ C | x - y | ^ {m - n } \ \ \textrm{ if } m < n,$$

$$| G ( x, y) | \leq C + C | \mathop{\rm ln} | x - y | | \ \textrm{ if } m = n,$$

are valid for the Green function, and the latter is uniformly bounded if $m > n$.

The boundary eigen value problem $Au = \lambda u$ is equivalent to the integral equation

$$u ( x) = \ \lambda \int\limits _ \Omega G ( x, y) u ( y) dy,$$

to which Fredholm's theory (cf. ) is applicable (cf. Fredholm theorems). Here, the Green function of the adjoint boundary value problem is $\overline{ {G ( y, x) }}\;$. It follows, in particular, that the number of eigen values is at most countable, and there are no finite limit points; the adjoint boundary value problem has complex-conjugate eigen values of the same multiplicity.

A Green function has been more thoroughly studied for second-order equations, since the nature of the singularity of the fundamental solution can be explicitly written out. Thus, for the Laplace operator the Green function has the form

$$G ( x, y) = \ - \frac{\Gamma ( n / 2) }{2 \pi ^ {n/2} ( n - 2) } | x - y | ^ {2 - n } + \gamma ( x, y) \ \ \textrm{ if } n > 2,$$

$$G ( x, y) = + \frac{1}{2 \pi } \mathop{\rm ln} | x - y | + \gamma ( x, y) \ \textrm{ if } n = 2,$$

where $\gamma ( x, y)$ is a harmonic function in $\Omega$ chosen so that the Green function satisfies the boundary condition.

The Green function $G ( x, y)$ of the first boundary value problem for a second-order elliptic operator $a ( x, D)$ with smooth coefficients in a domain $\Omega$ with Lyapunov-type boundary $\partial \Omega$, makes it possible to express the solution of the problem

$$a ( x, D) u( x) = \ f ( x) \ \textrm{ if } \ \ x \in \Omega ,\ \ \left . u \right | _ {\partial \Omega } = \phi ,$$

in the form

$$u ( x) = \ \int\limits _ \Omega G ( x, y) f ( y) dy + \int\limits _ {\partial \Omega } \frac \partial {\partial \nu _ {y} } G ( x, y) \phi ( y) d \sigma _ {y} ,$$

where $\partial / \partial \nu _ {y}$ is the derivative along the outward co-normal of the operator $a ( x, D)$ and $d \sigma _ {y}$ is the surface element on $\partial \Omega$.

If the homogeneous boundary condition $Au = 0$ has non-trivial solutions, a generalized Green function is introduced, just as for ordinary differential equations. Thus, a generalized Green function, the so-called Neumann function , is available for the Laplace operator.

2) Parabolic equations. Let $P$ be the parabolic differential operator of order $m$ generated by the differential polynomial

$$p \left ( x, t, D _ {x} ,\ \frac \partial {\partial t } \right ) = \ \frac \partial {\partial t } - \sum _ {| \alpha | \leq m } a _ \alpha ( x, t) D _ {x} ^ \alpha ,$$

$$x \in \Omega ,\ t > 0,$$

and the homogeneous initial and boundary conditions

$$u ( x, 0) = 0,\ \ B _ {j} u ( x, t) = 0,$$

where $B _ {j}$ are boundary operators with coefficients defined for $x \in \partial \Omega$ and $t \geq 0$. The Green function of the operator $P$ is a function $G ( x, t, y, \tau )$ which for arbitrary fixed $( y , \tau )$ with $t > \tau \geq 0$ and $y \in \Omega$ satisfies the homogeneous boundary conditions $B _ {j} = 0$ and also satisfies the equation

$$p \left ( x, t, D _ {x} ,\ \frac \partial {\partial t } \right ) G ( x, t, y, \tau ) = \ \delta ( x - y , t - \tau ) .$$

For operators with smooth coefficients and normal boundary conditions, which ensures the uniqueness of the solution of the problem $pu = 0$, a Green function exists, and the solution of the equation

$$p \left ( x, t, D _ {x} ,\ \frac \partial {\partial t } \right ) u ( x, t) = \ f ( x, t)$$

satisfying the homogeneous boundary conditions and the initial conditions $u ( x, 0) = \phi ( x)$, has the form

$$u ( x, t) = \ \int\limits _ { 0 } ^ { t } \ d \tau \int\limits _ \Omega G ( x, t, y, \tau ) f ( y, \tau ) dy +$$

$$+ \int\limits _ \Omega G ( x, t, y, 0) \phi ( y) dy.$$

In the study of elliptic or parabolic systems the Green function is replaced by the concept of a Green matrix, by means of which solutions of boundary value problems with homogeneous boundary conditions for these systems are expressed as integrals of the products of a Green matrix by the vectors of the right-hand sides and the initial conditions .

Green functions are named after G. Green (1828), who was the first to study a special case of such functions in his studies on potential theory.

How to Cite This Entry:
Green function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Green_function&oldid=47137
This article was adapted from an original article by Sh.A. AlimovV.A. Il'in (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article