Harmonic function

A real-valued function $ u $, defined in a domain $ D $ of a Euclidean space $ \mathbf R ^ {n} $, $ n \geq 2 $, having continuous partial derivatives of the first and second orders in $ D $, and which is a solution of the Laplace equation

$$ \Delta u \equiv \ \frac{\partial ^ {2} u }{\partial x _ {1} ^ {2} } + \dots + \frac{\partial ^ {2} u }{\partial x _ {n} ^ {2} } = 0, $$

where $ x _ {1} \dots x _ {n} $ are the orthogonal Cartesian coordinates of the point $ x $. This definition is sometimes extended to include complex functions $ w( x) = u( x) + iv( x) $ as well, in the sense that their real and imaginary parts $ \mathop{\rm Re} w( x) = u ( x) $ and $ \mathop{\rm Im} w ( x) = v ( x) $ are harmonic functions. The requirements of continuity and even of the existence of derivatives are not a priori indispensable. For instance, one of Privalov's theorems is applicable: A continuous function $ u $ in $ D $ is a harmonic function if and only if at any point $ x \in D $ the mean-value property

$$ u ( x) = \ { \frac{1}{\omega _ {n} ( R) } } \int\limits _ {B _ {n} ( x, R) } u ( y) dy $$

— where $ B _ {n} ( x, R) $ is the ball of radius $ R $ with centre at $ x $, $ \omega _ {n} ( R) $ is the volume of this ball and $ dy $ is the volume element in $ \mathbf R ^ {n} $— is fulfilled for sufficiently small $ R > 0 $.

If $ D $ is unbounded with a compact boundary $ \partial D $, the definition of a harmonic function may be completed to include the point at infinity $ \infty $, i.e. it may additionally be defined in domains in the Aleksandrov compactification of $ \mathbf R ^ {n} $. The general principle of such a completion of the definition is that, under the simplest transformations which preserve harmonicity (inversion if $ n = 2 $, Kelvin transformations if $ n \geq 3 $, cf. Kelvin transformation) and map a finite point $ x _ {0} $ into $ \infty $, a harmonic function in a neighbourhood of $ x _ {0} $ becomes a harmonic function in a neighbourhood of $ \infty $. On this basis, a harmonic function $ u $ is said to be regular at infinity for $ n \geq 3 $ if

$$ \lim\limits _ {| x | \rightarrow \infty } \ u ( x) = 0,\ \ | x | = \sqrt {x _ {1} ^ {2} + \dots + x _ {n} ^ {2} } . $$

Thus, for a harmonic function $ u $ which is regular at infinity one always has $ u( \infty ) = 0 $ if $ n \geq 3 $. If $ n = 2 $, the condition

$$ u ( x) = O( 1),\ | x | \rightarrow \infty , $$

which implies the existence of a finite limit

$$ \lim\limits _ {| x | \rightarrow \infty } \ u ( x) = u ( \infty ), $$

must be met. Harmonic functions in unbounded domains are usually understood to mean harmonic functions regular at infinity.

In the theory of harmonic functions an important role is played by the principal fundamental solutions of the Laplace equation:

$$ h _ {2} ( x) = \ { \frac{1}{2 \pi } } \ \mathop{\rm ln} { \frac{1}{| x | } } \ \ \textrm{ if } \ n = 2, $$

$$ h _ {n} ( x) = { \frac{1}{( n - 2) \sigma _ {n} } } { \frac{1}{| x | ^ {n - 2 } } } \ \textrm{ if } n \geq 3, $$

where $ \sigma _ {n} $ is the surface area of the unit sphere in $ \mathbf R ^ {n} $. If $ | x | > 0 $, this is a harmonic function. The fundamental solutions may be used to write down the basic formula of the theory of harmonic functions, which expresses the values of a harmonic function $ u ( x) $ inside a domain $ D $ in terms of its values $ u( y) $ on the boundary $ S = \partial D $ and the values of its derivative in the direction of the exterior normal $ \partial u ( y) / \partial \nu $ towards $ S $ at the point $ y $:

$$ \int\limits _ { S } \left [ h _ {n} ( x - y) \frac{\partial u ( y) }{\partial \nu } - u ( y) \frac{\partial h _ {n} ( x - y) }{\partial \nu _ {y} } \right ] d \sigma ( y) = $$

$$ = \ \left \{ \begin{array}{ll} u ( x), & x \in D, \\ 0, & x \notin \overline{D}\; . \\ \end{array} \right .$$

This Green formula is valid, for example, if the function $ u $ and its partial derivatives of the first order are continuous in the closed domain $ \overline{D}\; $, i.e. if $ u \in C ^ {1} ( \overline{D}\; ) $, the boundary $ S $ of which is a piecewise-smooth closed surface or curve. It yields a representation of an arbitrary harmonic function $ u $ as the sum of single- and double-layer potentials (cf. Potential theory). The densities of these potentials, i.e. the boundary values $ \partial u ( y) / \partial \nu $ and $ u( y) $ respectively, cannot be specified arbitrarily. There is an integral relationship between the two, in that the left-hand side of the last-named formula — the Green integral — must vanish for all points $ x $ outside $ \overline{D}\; $. The basic formula of the theory of harmonic functions is a direct analogue of the fundamental formula of the theory of analytic functions — the integral formula of Cauchy (cf. Cauchy integral). This formula also remains valid if the principal fundamental solution $ h _ {n} $ in it is replaced by any other fundamental solution of the Laplace equation which is sufficiently smooth in $ \overline{D}\; $, e.g. belongs to $ C ^ {1} ( \overline{D}\; ) $.

The fundamental properties of harmonic functions, on the assumption that the boundary $ S $ of the domain $ D $ is piecewise smooth, are listed below. After suitable modification, many of them are also valid for complex harmonic functions.

1) If $ D $ is a bounded domain and a harmonic function $ u \in C ^ {1} ( \overline{D}\; ) $, then

$$ \int\limits _ { S } \frac{\partial u ( y) }{\partial \nu } \ d \sigma ( y) = 0. $$

2) The mean-value theorem: If $ u $ is a harmonic function in the ball $ B = B ( x _ {0,\ } R) $ of radius $ R $ with centre at $ x _ {0} $ and if $ u \in C ^ {1} ( \overline{B}\; ) $, then its value at the centre of the ball is equal to the value of its arithmetical mean on the sphere $ S ( x _ {0} , R) $, i.e.

$$ u ( x _ {0} ) = \ { \frac{1}{\sigma _ {n} ( R) } } \int\limits _ {S ( x _ {0} , R) } u ( y) d \sigma ( y), $$

where $ \sigma _ {n} ( R) $ is the surface area of the sphere of radius $ R $ in $ \mathbf R ^ {n} $. If $ u $ is only continuous, this property may be taken as the definition of a harmonic function.

3) The maximum/minimum principle: Let $ D $ be a domain in $ \mathbf R ^ {n} $ not containing $ \infty $ as an interior point. If $ u $ is a harmonic function in $ D $, and $ u( x) \neq \textrm{ const } $, then $ u $ cannot attain a local extremum at any point $ x _ {0} \in D $, i.e. in any neighbourhood $ V ( x _ {0} ) $ of any point $ x _ {0} \in D $ there exists a point $ x ^ {*} \in V( x _ {0} ) $ at which $ u ( x ^ {*} ) > u ( x _ {0} ) $ and there exists a point $ x ^ {*} \in V ( x _ {0} ) $ at which $ u ( x ^ {*} ) < u ( x _ {0} ) $( the maximum/minimum principle in local form). If, in addition, $ u \in C ( \overline{D}\; ) $, then the largest and the least values of $ u $ on $ \overline{D}\; $ are attained only at the points of the boundary $ \partial D $( the maximum/minimum principle in global form). Consequently, if $ | u( x) | \leq M $ on $ \partial D $, then $ | u( x) | \leq M $ everywhere in $ \overline{D}\; $.

This principle may be generalized in various ways.

For instance, if $ u $ is a harmonic function in a domain $ D $ not containing $ \infty $ and if

$$ \lim\limits _ {x \rightarrow y } \ \sup u ( x) \leq M $$

for all points $ y \in \partial D $( boundary in $ \overline{ {\mathbf R ^ {n} }}\; $), then $ u( x) \leq M $ everywhere in $ D $.

4) The theorem on removable singularities: If $ u $ is a harmonic function in a domain $ D \setminus \{ x _ {0} \} $, $ x _ {0} \in D $, which satisfies the condition

$$ u ( x) = o ( | h _ {n} ( x - x _ {0} ) | ),\ \ x \rightarrow x _ {0} , $$

then there exists a finite limit

$$ \lim\limits _ {x \rightarrow x _ {0} } \ u ( x) = u ( x _ {0} ) , $$

and $ u $ completed by the value $ u( x _ {0} ) $ is a harmonic function in $ D $.

5) If $ u $ is a harmonic function throughout the space $ \mathbf R ^ {n} $, $ n \geq 2 $, bounded from above or from below, then $ u = \textrm{ const } $.

6) If $ u $ is a harmonic function in a neighbourhood of a point $ x _ {0} = (( x _ {1} ) _ {0} \dots ( x _ {n} ) _ {0} ) $, then $ u $ can be expanded in this neighbourhood into a power series in the variables $ x _ {1} - ( x _ {1} ) _ {0} \dots x _ {n} - ( x _ {n} ) _ {0} $, i.e. all harmonic functions are analytic functions of the variables $ x _ {1} \dots x _ {n} $; consequently, a harmonic function $ u $ has derivatives of all orders:

$$ \frac{\partial ^ {m} u }{\partial x _ {1} ^ {k _ {1} } \dots \partial x _ {n} ^ {k _ {n} } } ,\ \ k _ {1} + \dots + k _ {n} = m, $$

which are also harmonic functions.

7) The uniqueness property: If $ u $ is a harmonic function in a domain $ D \subset \mathbf R ^ {n} $ and $ u \equiv 0 $ in some $ n $- dimensional neighbourhood of some point $ x _ {0} \in D $, then $ u \equiv 0 $ in $ D $. If $ u $ is an analytic function of the real variables $ x = ( x _ {1} \dots x _ {n} ) $ in a domain $ D \subset \mathbf R ^ {n} $ and if $ u $ is a harmonic function in some $ n $- dimensional neighbourhood of an arbitrary point $ x _ {0} \in D $, then $ u $ is a harmonic function in $ D $.

8) The symmetry principle: Let the boundary $ \partial D $ of a domain $ D \subset \mathbf R ^ {n} $ contain a set $ G $ that is open in the plane $ x _ {n} = 0 $, let $ u $ be a harmonic function in $ D $ such that $ u = 0 $ and continuous on $ G $ and let $ \widetilde{D} $ be the domain symmetrical to $ D $ with respect to the plane $ x _ {n} = 0 $; then $ u $ can be harmonically extended into the domain $ D \cup G \cup \widetilde{D} $ by the formula

$$ u ( x _ {1} \dots x _ {n - 1 } , x _ {n} ) = \ - u ( x _ {1} \dots x _ {n - 1 } , - x _ {n} ), $$

$$ ( x _ {1} \dots x _ {n - 1 } , x _ {n} ) \in \widetilde{D} . $$

9) Harnack's first theorem: If a sequence $ \{ u _ {n} \} $ of harmonic functions in a bounded domain $ D $, continuous in the closed domain $ \overline{D}\; $, converges uniformly on the boundary $ \partial D $, then it converges uniformly on $ \overline{D}\; $, and the limit function

$$ u ( x) = \ \lim\limits _ {n \rightarrow \infty } \ u _ {n} ( x) $$

is a harmonic function in $ D $.

10) Harnack's second theorem: If a sequence $ \{ u _ {n} \} $ of harmonic functions is monotone in a domain $ D $ and converges at least at one point $ x _ {0} \in D $, then it converges everywhere in $ D $ towards a harmonic function

$$ u ( x) = \ \lim\limits _ {n \rightarrow \infty } \ u _ {n} ( x). $$

See also Harnack inequality; Harnack theorem.

There exists a close connection between harmonic functions of two variables $ ( x _ {1} , x _ {2} ) $ and analytic functions of the complex variable $ z = x _ {1} + i x _ {2} $. The real and the imaginary part of an analytic function are, possibly multi-valued, conjugate harmonic functions, i.e. they are connected by the Cauchy-Riemann equations. If a harmonic function $ u ( x _ {1} , x _ {2} ) $ is defined in a neighbourhood of a point $ ( x _ {1} ^ {0} , x _ {2} ^ {0} ) $, the simplest way of finding an analytic function $ f( z) $, $ z = x _ {1} + i x _ {2} $, for which $ u ( x _ {1} , x _ {2} ) = \mathop{\rm Re} f( z) $ is given by the Goursat formula:

$$ f ( z) = 2u \left ( \frac{z + \overline{ {z ^ {0} }}\; }{2 } ,\ \frac{z - \overline{ {z ^ {0} }}\; }{2i } \right ) - u ( x _ {1} ^ {0} , x _ {2} ^ {0} ) + iC _ {0} , $$

where $ \overline{ {z ^ {0} }}\; = x _ {1} ^ {0} - i x _ {2} ^ {0} $ and $ C _ {0} $ is an arbitrary real constant. Certain spatial problems in mathematical physics also involve multi-valued harmonic functions in domains in $ \mathbf R ^ {n} $, $ n \geq 2 $.

The major importance of harmonic functions in mathematical physics is mainly due to the frequent occurrence of vector fields of the form $ \mathbf s = - \mathop{\rm grad} u $. Such fields in domains not containing field sources must satisfy the conservation equation $ \mathop{\rm div} \mathbf s = - \Delta u = 0 $, i.e. the Laplace equation, which means that in such domains $ u $ is a harmonic function.

Examples. If $ \mathbf s $ is the force vector of the gravity field, $ u $ is the Newton potential of the gravitational forces; if $ \mathbf s $ is the field of velocities of a stationary motion of an incompressible homogeneous gas or liquid, $ u $ is the velocity potential; if $ \mathbf s $ is the strength of an electrostatic field in a homogeneous isotropic medium, $ u $ is the potential of the electrostatic field; if $ \mathbf s $ is the strength of a stationary magnetic field in a homogeneous isotropic medium, $ u $ is the scalar, usually multi-valued, potential of the magnetic field. In the case of steady propagation of heat in a uniform isotropic medium or a stationary distribution of diffusing particles, the harmonic function $ u( x) $ is the temperature of the medium or the density of the particles at a point $ x $, respectively. Many important problems in the theory of elasticity and in the theory of electromagnetic fields can also be reduced to solving problems concerning harmonic functions.

The boundary Dirichlet problem, or the first boundary value problem, is of special importance in the development of the theory of harmonic functions and mathematical physics. It consists in finding a function $ u $ which is harmonic in a domain $ D $ and continuous on $ \overline{D}\; $, from given continuous values $ u( y) $ on the boundary $ S = \partial D $. If the surface or line $ S $ is sufficiently smooth, the solution may be expressed by the Green function $ G( x, y) $:

$$ u ( x) = \ - \int\limits _ { S } u ( y) \frac{\partial G ( x, y) }{\partial \nu _ {y} } \ d \sigma ( y). $$

In the case of the simplest domains (spheres, half-spaces), when the normal derivative is readily expressed in explicit form, the Poisson integral is obtained. The second boundary value problem, or the Neumann problem, is also often encountered. It consists in determining a harmonic function $ u $ from given values of its normal derivative on the boundary $ S $. This problem can be solved using the corresponding Green function, but explicit expressions are much more complicated in this case. There are many more boundary value problems in the theory of harmonic functions, the formulations and solutions of which are more complicated. See also Balayage method; Robin problem.

A special place in the modern theory of harmonic functions is occupied by ill-posed problems, mainly those connected with the Cauchy problem for the Laplace equation. These include, for example, the following problem on the best majorant: If on the boundary $ S = \partial D $ of a domain $ D $ the function $ M = M( y) $ and the conditions $ | u( y) | \leq M ( y) $, $ | \partial u ( y) / \partial \nu | \leq M ( y) $ are given, find the best possible estimate of $ \sup | u( x) | $ in the class of harmonic functions $ u $ in $ D $[9], [10].

The study of the boundary properties of harmonic functions related with subharmonic functions (cf. Subharmonic function) and with the boundary properties of analytic functions is of importance. For instance, a harmonic function $ u $ in the unit ball $ B ( 0, 1 ) $ of $ \mathbf R ^ {n} $ usually has no radial limit values

$$ f ( y) = \ \lim\limits _ {r \uparrow 1 } u ( ry),\ \ y \in S = \partial B ( 0, 1). $$

However, in the case of the class $ A $ of harmonic functions defined by the condition

$$ \int\limits _ { S } u ^ {+} ( ry) d \sigma ( y) \leq \ C ( u) < \infty , $$

where $ d \sigma ( y) $ is the surface element of $ S $, $ u ^ {+} = \max \{ 0, u \} $, the radial boundary values exist almost-everywhere on $ S $ with respect to Lebesgue measure, and an $ u \in A $ can be represented in the form of a Poisson–Stieltjes integral

$$ u ( x) = \int\limits _ { S } P _ {n} ( x, y) \ d \mu ( y), $$

where

$$ P _ {n} ( x, y) = \ { \frac{1}{\sigma _ {n} ( 1) } } \frac{1 - | x | ^ {2} }{| x - y | ^ {n} } $$

is the Poisson kernel and $ d \mu $ is the Borel measure on $ S $. The proper subclass $ B $ of the class $ A $ consisting of all harmonic functions $ u $ that can be represented in $ B ( 0, 1) $ by a Poisson–Lebesgue integral,

$$ u ( x) = \int\limits _ { S } P _ {n} ( x, y) f ( y) d \sigma ( y), $$

is also of importance.

Substantial advances have been made in the axiomatic theory of harmonic functions and potentials in topological spaces (cf. Harmonic space; Potential theory, abstract).

References

[1]	A.F. Timan, V.N. Trofimov, "Introduction to the theory of harmonic functions" , Moscow (1968) (In Russian) MR0315142
[2]	N.M. [N.M. Gyunter] Günter, "Potential theory and its applications to basic problems of mathematical physics" , F. Ungar (1967) (Translated from Russian) MR0222316 Zbl 0164.41901
[3]	L.N. Sretenskii, "Theory of the Newton potential" , Moscow-Leningrad (1946) (In Russian)
[4]	M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959) MR0106366 Zbl 0084.30903
[5]	O.D. Kellogg, "Foundations of potential theory" , Springer (1967) (Re-issue: Springer, 1967) MR0222317 Zbl 0152.31301
[6]	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
[7]	M.A. Lavrent'ev, B.V. Shabat, "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft. (1967) (Translated from Russian)
[8]	A.I. Markushevich, "Theory of functions of a complex variable" , 2 , Chelsea (1977) (Translated from Russian) MR0444912 Zbl 0357.30002
[9]	M.M. [M.M. Lavrent'ev] Lavrentiev, "Some improperly posed problems of mathematical physics" , Springer (1967) (Translated from Russian) Zbl 0149.41902
[10]	S.N. Mergelyan, "Harmonic approximation and approximate solution of Cauchy's problem for the Laplace equation" Uspekhi Mat. Nauk , 11 : 5 (1956) pp. 3–26 (In Russian)
[11]	I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian) MR0083565
[12]	E.D. Solomentsev, "Harmonic and subharmonic functions and their generalizations" Itogi Nauk. Ser. Mat., Mat. Anal., Teor. Veroyatnost., Regulirovanie, 1962 (1964) pp. 83–100 (In Russian)

Comments

Fundamental property 5) above is also called Picard's theorem (for harmonic functions).

The symmetry principle is also known as the Schwarz reflection principle (cf. Schwarz symmetry theorem). [a1] defines harmonic functions with respect to the fine topology.

References

[a1]	B. Fuglede, "Finely harmonic functions" , Springer (1972) MR0450590 Zbl 0248.31010
[a2]	W.K. Hayman, P.B. Kennedy, "Subharmonic functions" , 1 , Acad. Press (1976) MR0460672 MR0419791 MR0412442 MR0442324 Zbl 0419.31001 Zbl 0339.31003 Zbl 0328.33011
[a3]	L.L. Helms, "Introduction to potential theory" , Wiley (Interscience) (1969)