Green formulas

Formulas of the integral calculus of functions of several variables, connecting the values of an $ n $- fold integral over a domain $ D $ in an $ n $- dimensional Euclidean space $ E ^ {n} $ with an $ ( n - 1) $- fold integral along the piecewise-smooth boundary $ \partial D = \Gamma $ of this domain. The Green formulas are obtained by integration by parts of integrals of the divergence of a vector field that is continuous in $ \overline{D}\; = D + \Gamma $ and that is continuously differentiable in $ D $.

In the simplest Green formula,

$$ \tag{1 } \int\limits _ \Gamma [ P dx + Q dy] = \ \int\limits \int\limits _ { D } \left [ \frac{\partial Q }{\partial x } - \frac{\partial P }{\partial y } \right ] dx dy, $$

the curvilinear integral along the contour $ \Gamma $ is expressed as a double integral over the domain $ D \subset E ^ {2} $. Here the domain $ D $ is oriented in a natural manner, while an induced orientation, known as counterclockwise, is taken along the boundary $ \Gamma $. Formula (1) has a simple hydrodynamic meaning: The flow across the boundary $ \Gamma $ of a liquid flowing on a plane at rate $ \mathbf v = ( Q, - P) $ is equal to the integral over $ D $ of the intensity (divergence) $ \mathop{\rm div} \mathbf v = ( \partial Q/ \partial x) - ( \partial P/ \partial y) $ of the sources and sinks distributed over $ D $. In this sense the Green formula (1) resembles the Ostrogradski formula (see also Stokes formula).

Formula (1) is sometimes attributed to C.F. Gauss and B. Riemann, but none of its usual appellations corresponds to historical truth; it was in fact encountered as early as the 18th century in the analytical studies of L. Euler and others.

G. Green [1] must be credited with the following formulas of potential theory:

$$ \tag{2 } \int\limits _ { D } \left [ v \Delta u + \sum _ {i = 1 } ^ { 3 } \frac{\partial u }{\partial x ^ {i} } \frac{\partial v }{\partial x ^ {i} } \right ] dx = \ \int\limits _ \Gamma v \frac{\partial u }{\partial N } ds, $$

the preparatory Green formula, and

$$ \tag{3 } \int\limits _ { D } [ u \Delta v - v \Delta u] dx = \ \int\limits _ \Gamma \left [ u \frac{\partial v }{\partial N } - v \frac{\partial u }{\partial N } \right ] ds. $$

Here $ D $ is a domain in $ E ^ {3} $, $ x = ( x ^ {1} , x ^ {2} , x ^ {3} ) $, $ dx = dx ^ {1} dx ^ {2} dx ^ {3} $ is the volume element of $ D $, $ ds $ is the surface element of $ \Gamma $, $ N = ( N _ {1} , N _ {2} , N _ {3} ) $ is the unit outer (co-)normal to $ \Gamma $,

$$ \frac \partial {\partial N } = \ \sum _ {i = 1 } ^ { 3 } N _ {i} \frac \partial {\partial x ^ {i} } $$

is the operator of differentiation in the direction of the (co-)vector $ N $, and

$$ \Delta = \ \sum _ {i = 1 } ^ { 3 } \left ( \frac \partial {\partial x ^ {i} } \right ) ^ {2} $$

is the Laplace operator.

Formulas (2) and (3) are also valid if $ D $ is a domain in $ E ^ {n} $, $ x = ( x ^ {1} \dots x ^ {n} ) $, $ dx = dx ^ {1} \dots dx ^ {n} $ is the volume element in $ D $, $ ds $ is the $ ( n - 1) $- dimensional volume element of $ \Gamma $, and

$$ \Delta = \ \sum _ {i = 1 } ^ { n } \left ( \frac \partial {\partial x ^ {i} } \right ) ^ {2} $$

is the Laplace operator in $ n $ independent variables.

The generalizations of the Green formulas (2) and (3) for linear partial differential operators with sufficiently smooth coefficients have the following form:

1) If

$$ Lu = \ \sum _ {i, j = 1 } ^ { n } \frac \partial {\partial x ^ {i} } \left [ a ^ {ij} ( x) \frac{\partial u }{\partial x ^ {j} } \right ] + \sum _ {i = 1 } ^ { n } b ^ {i} ( x) \frac{\partial u }{\partial x ^ {i} } + c ( x) u , $$

$$ L ^ {*} v = \sum _ {i, j = 1 } ^ { n } \frac \partial {\partial x ^ {i} } \left [ a ^ {ij} ( x) \frac{\partial v }{\partial x ^ {j} } \right ] - \sum _ {i = 1 } ^ { n } \frac \partial {\partial x ^ {i} } [ b ^ {i} ( x) v] + c ( x) v $$

are (real) adjoint second-order differential operators, $ a ^ {ij} = a ^ {ji} $, then

$$ \int\limits _ { D } \left [ vLu + \sum _ {i, j = 1 } ^ { n } a ^ {ij} \frac{\partial u }{\partial x ^ {i} } \frac{\partial v }{\partial x ^ {j} } - \left ( \sum _ {i = 1 } ^ { n } b ^ {i} \frac{\partial u }{\partial x ^ {i} } + cu \right ) v \right ] dx = $$

$$ = \ \int\limits _ \Gamma v \frac{\partial u }{\partial M } ds, $$

$$ \int\limits _ { D } [ uL ^ {*} v - vLu] dx = \ \int\limits _ \Gamma \left [ u \frac{\partial v }{\partial M } - v \frac{\partial u }{\partial M } - B uv \right ] ds, $$

where $ N = ( N _ {1} \dots N _ {n} ) $ is the unit (co-)vector of the outer normal to $ \Gamma $:

$$ B = \ \sum _ {i = 1 } ^ { n } N _ {i} b ^ {i} ,\ \ \frac \partial {\partial M } = \ \sum _ {i, j = 1 } ^ { n } a ^ {ij} N _ {j} \frac \partial {\partial x ^ {i} } $$

is the operator of differentiation in the direction of the so-called co-normal

$$ M = \left ( \sum _ {j = 1 } ^ { n } a ^ {1j} N _ {j} \dots \sum _ {j = 1 } ^ { n } a ^ {nj} N _ {j} \right ) $$

of the operator $ L $.

2) If

$$ Lu = \ \sum _ {i, j = 1 } ^ { n } a ^ {ij} ( x) \frac{\partial ^ {2} u }{\partial x ^ {i} \partial x ^ {j} } + \sum _ {i = 1 } ^ { n } b ^ {i} ( x) \frac{\partial u }{\partial x ^ {i} } + c ( x) u , $$

$$ L ^ {*} v = \sum _ {i, j = 1 } ^ { n } \frac{\partial ^ {2} }{\partial x ^ {i} \partial x ^ {j} } [ a ^ {ij} ( x) v] - \sum _ {i = 1 } ^ { n } \frac \partial {\partial x ^ {i} } [ b ^ {i} ( x) v] + c ( x) v, $$

then

$$ \int\limits _ { D } \left [ vLu + \sum _ {i, j = 1 } ^ { n } a ^ {ij} \frac{\partial u }{\partial x ^ {i} } \frac{\partial v }{\partial x ^ {j} } - \left ( \sum _ {i = 1 } ^ { n } b ^ {i} \frac{\partial u }{\partial x ^ {i} } + cu \right ) v \right ] = $$

$$ = \ \int\limits _ \Gamma v \frac{\partial u }{\partial M } ds, $$

where $ M $ is the co-normal of $ L $, and

$$ C = \ \sum _ {i = 1 } ^ { n } \left [ b ^ {i} - \sum _ {j = 1 } ^ { n } \frac{\partial a ^ {ij} }{\partial x ^ {j} } \right ] N _ {i} . $$

3) If

$$ Lu = \ \sum _ {p = 1 } ^ { m } \ \sum _ {| \alpha | = p } a ^ \alpha ( x) D _ \alpha u + a ( x) u , $$

$$ L ^ {*} v = \sum _ {p = 1 } ^ { m } (- 1) ^ {p} D _ \alpha [ a ^ \alpha ( x) v] + a ( x) v $$

are (real) adjoint differential operators of order $ m $, $ \alpha = ( \alpha _ {1} \dots \alpha _ {p} ) $ is an integer multi-index of length $ | \alpha | = p $, $ 1 \leq \alpha _ {i} \leq n $, $ D _ \alpha = \partial _ {\alpha _ {1} } \dots \partial _ {\alpha _ {p} } $, and $ \partial _ {i} = \partial / \partial x ^ {i} $, then

$$ \tag{4 } \int\limits _ { D } [ uL ^ {*} v - vLu] dx = $$

$$ = \ \sum _ {p = 1 } ^ { m } \sum _ {| \alpha | = p } \sum _ {k = 1 } ^ { p } \int\limits _ \Gamma (- 1) ^ {k} \left [ \partial _ {\alpha _ {1} } \dots \partial _ {\alpha _ {k - 1 } } ( a ^ \alpha v) \right ] \times $$

$$ \times N _ {\alpha _ {k} } \left [ \partial _ {\alpha sub {k + 1 } } \dots \partial _ {\alpha _ {p} } u \right ] ds. $$

Here the boundary integral can be written as the bilinear sum

$$ \sum _ { ij } \int\limits _ \Gamma B ^ {ij} ( S _ {i} u) ( T _ {j} v) ds, $$

where $ S _ {i} $, $ T _ {j} $ are certain linear differential operators of orders $ s _ {i} $, $ t _ {j} $, $ 0 \leq s _ {i} + t _ {j} \leq m - 1 $.

Green's formulas play an important role in analysis and, particularly, in the theory of boundary value problems for differential operators (both ordinary and partial differential operators) of the second or higher orders. For functions $ u $, $ v $ which are sufficiently smooth in $ \overline{D}\; $, Green's formulas (2) and (4) serve as the source of several relations which are useful in the study of solutions of boundary value problems, in the clarification of the type of boundary value problems, in obtaining explicit solutions, etc. Thus, the function $ u $ in (2), which is harmonic in $ D $, satisfies the Gauss theorem

$$ \int\limits _ {\partial G } \frac{\partial u }{\partial N } \ ds = 0 $$

for $ v \equiv 1 $. For sufficiently smooth functions $ u $, $ w $ in $ \overline{D}\; $, and the function

$$ v ( x) = \ \left \{ \begin{array}{ll} \frac{1}{n - 2 } | x - y | ^ {2 - n } + w ( x) & \textrm{ if } n \geq 3, \\ - \mathop{\rm ln} | x - y | + w ( x) & \textrm{ if } n = 2, \\ \end{array} \right .$$

which, for $ x = y $, has the same singularity as the fundamental solution of the Laplace operator, the following Green formulas are valid:

$$ \tag{5 } \int\limits _ { D } \left [ u \Delta w + \sum _ {i = 1 } ^ { n } \frac{\partial u }{\partial x ^ {i} } \frac{\partial v }{\partial x ^ {i} } \right ] dx = \ cu ( y) + \int\limits _ { D } u \frac{\partial v }{\partial N } ds, $$

$$ \tag{6 } \int\limits _ { D } [ u \Delta w - v \Delta u] dx = cu ( y) + \int\limits _ { D } \left [ u \frac{\partial v }{\partial N } - v \frac{\partial u }{\partial N } \right ] ds. $$

Here,

$$ c = \left \{ \begin{array}{ll} \omega _ {n} & \textrm{ if } y \in D, \\ { \frac{1}{2} } \omega _ {n} & \textrm{ if } y \in \Gamma , \\ 0 & \textrm{ if } y \notin \overline{D}\; , \\ \end{array} \right .$$

where $ \omega _ {n} = 2 \pi ^ {n/2} / \Gamma ( n/2) $ is the area of the $ ( n - 1) $- dimensional unit sphere in $ E ^ {n} $. Here it is assumed, for $ y \in \Gamma $, that the boundary $ \Gamma $ has a continuous tangent plane in some neighbourhood of $ y $.

Formulas (5) and (6) serve to obtain integral representations of the solutions of basic boundary value problems in potential theory (cf. Harmonic function; Green function; Poisson formula). Thus, they are used to obtain the Green formula, or Green integral,

$$ \tag{7 } u ( y) = \ { \frac{1}{c} } \int\limits _ \Gamma \left [ v ( x) \frac{\partial u }{\partial N } - u ( x) \frac{\partial v }{\partial N } \right ] ds, $$

for a function $ u $ which is harmonic in $ D $. This integral plays an important role in the theory of harmonic functions (cf. Harmonic function). Formulas resembling (5) and (6), which give integral representations of the solution of the Cauchy problem or of the mixed problem, are also valid for a normal hyperbolic operator of order two. See Kirchhoff formula; Riemann method; Riemann function. For Green formulas in the theory of boundary value problems see also [4]–[9].

References

[1]	G. Green, "An essay on the application of mathematical analysis to the theories of electricity and magnetism" , Nottingham (1828) (Reprint: Mathematical papers, Chelsea, reprint, 1970, pp. 1–82) Zbl 21.0014.03
[2]	J. Maxwell, "Selected works on the theory of electromagnetic fields" , Moscow (1954) (In Russian; translated from English)
[3]	V.I. Smirnov, "A course of higher mathematics" , 2 , Addison-Wesley (1964) (Translated from Russian) MR0182690 MR0182688 MR0182687 MR0177069 MR0168707 Zbl 0122.29703 Zbl 0121.25904 Zbl 0118.28402 Zbl 0117.03404
[4]	R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German) MR0195654
[5]	V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian) MR0764399 Zbl 0954.35001 Zbl 0652.35002 Zbl 0695.35001 Zbl 0699.35005 Zbl 0607.35001 Zbl 0506.35001 Zbl 0223.35002 Zbl 0231.35002 Zbl 0207.09101
[6]	S.L. Sobolev, "Partial differential equations of mathematical physics" , Pergamon (1964) (Translated from Russian) MR0178220 Zbl 0123.06508
[7]	C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian) MR0284700 Zbl 0198.14101
[8]	N. Dunford, J.T. Schwartz, "Linear operators. Spectral theory" , 2 , Interscience (1963) MR0188745
[9]	J.L. Lions, E. Magenes, "Non-homogenous boundary value problems and applications" , 1–2 , Springer (1972) (Translated from French)