On the mean-value property for polyharmonic functions

In investigation of mathematical models described by the polyharmonic equation properties of polyharmonic functions are very useful to know. Let u ( x ) be a harmonic function in the domain Q C R n and B r ( x ) = {y E R ⁿ : |y — x| < r} . It is well known (see [1]) the Gauss mean-value property for harmonic functions: if x E Q and B r ( x ) C Q , then for all functions harmonic in Q

u ⁽ x ) = i^B ¹ ( x ) T дв г, ( . ) ^{u ( y 1}

This mean-value property has been extended by Pizzetti (see [2]) for k-harmonic functions in Q to the form r 2 гд iu ( x 0)

4 i i !Г( i + n/ 2) ^,

1      /                           X 1

dB(x)l двг,(.)u(y 1 dsy = r(n/2) X- where Г(a) is the Euler’s gamma function. This property can be easily written for a k-harmonic function u E CX1(Si') in the unit ball S C Rn in the form

₁                   X- 1

;J_№^u ( x )=g

Д i u (0) (2 , 2) г ( n, 2) г ,

where шп is the surface area of the unit sphere dS, and (a, b)x = a(a + b) • • • (a + b(k — 1)) is the generalized Pochhammer symbol with (a,b)o = 1. For example, (2, 2)г = (2i)!!. The similar formula was proved in [3, Theorem 7] for calculating the integral of homogeneous polynomial Qm (x) on the unit sphere j^ I   Qm (x) dsx

0 ,

*     Дm/2 Qm ( x )    ш m!! n ••• (n + m — 2) n m E 2N — 1

m E 2N

Consider the operator Л defined by the equality

n

^Л = X Xi Ь i =1

This operator plays an important role in our investigation because in the paper [4] it was proved that the following equality is fulfilled on ∂S

∂ ^k u ∂ν ^k

= Л ^[ k 1 u,

where v is the outer normal to dS , and t ^[ k ¹ = t ( t — 1) ... ( t — k + 1) is a factorial power of t . Besides, it is known (see, for example, [5]) that if u is a harmonic function, then function P (Л) u is also a harmonic one, where P ( A ) is a polynomial.

1.    The mean-value property for normal derivatives

We are going to extend formula (2) to the normal derivatives of the function u ( x ) . Let us denote N q = N U { 0 } .

Theorem 1. For all m G N q and for any polyharmonic function in the unit ball u G C ^m ( S ) the following equality holds

- [ "m^ds- = X , (2k)[mV A‘u(0), Шп-JdS dvm x k= (2,2)k(n, 2)k' '’

where ν is the unit outer normal to ∂S.

Proof. In [6, Theorem 4] it is proved that for any polyharmonic in S function u ( x ) the following Almansi representation takes place

^ 1 Ixl2 k Г1 (1 — a) k-1.

u(X)= vq(X) + 52 4k~kP j   (k - 1), an/ 1 Vk(ax) da,(6)

where harmonic in S functions v q( x ) ,..., V k ( x ) ,... are given by the formula

/ \ а к ( \    V'' (—1)s Ix|2s Z1 (1 — a)s 1 as 1 n/2—i л k+s /

V k ( x ) = A ^k u ( x ) + X^^SS T J --- ( _s - 1), ---a / 1 A ^{k +} u ( ax ) da. (7)

The upper limit of sum above is equal to infinity but since the function u ( x ) is a polyharmonic in S then summation is finite and exists k q such that V k ( x ) = 0 for all k > k q . It is not hard to see that

Л( |x|2 k u) = |x|2 k (2 k + Л) u and therefore

Л[2] (|x|2ku) = (Л — 1)(|x|2k(2k + Л)u) = |x|2k(2k — 1 + Л)(2k + Л)u, whence

Л[m 1 (|x|2ku) = |x|2k(2k — m + 1 + Л)... (2k — 1 + Л)(2k + Л)u = (2k)[m 1 u + Qm(Л)u, where Qm(A) is a certain polynomial such that Qm(0) = 0. Therefore in S we have

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ

Л ^[ m 1 u ( x ) = Л [ m^ v o ( x ) + X X ^2k 4 m ’ ! x ² k [^{1 (1} . - - 11 k ¹ a n ² - ¹ V k ( ax ) da + k =1              -                ⁽         )•

X IxT Г ¹ (1 - a)^{k- 1}    / 2 - 1

⁺ / v 4 k k ! y₀    ( k - 1)! ^a     Q m ^(Л) v k ⁽ )

Using the mean-value property for harmonic functions and harmonicity of Л V k we can obtain the equality J^ Q m (Л) V k ( ax ) ds x = 0 . Therefore using (7) we have

Л [ m u ( x ) ds_x = X            A1 - a ) k

¹ a n/ ^{2 1} dav k (0) =

“1-           x  k — 4 ^kк !( к - 1)! V ^J

_ X (2 к ) m v _k (0) _ X (2 к ) m A ^ku (0) = b(2 , 2) k ( n, 2) k =      (2 , 2) k ( n, 2) k ^.

k =1                  k =o

= X (2 к ) [ ^{m ]} V k (0) Г( к )Г( n/ 2) k —1 4 ^kк !( к — 1)! Г( к + n/ 2)

Hence, by virtue of (4), we obtain the theorem’s statement for m >  0 . If m = 0 , then by equality (2) the formula (5) is true in this case also.

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Example 1. Let function u ( x ) be a harmonic in S and u E C ^ ( S ) , then from Theorem 1 follows that

∂ ^m u

У , _s dv m ds x = ⁰ - ^m - ¹ •

For a biharmonic in S function u E C^ (S) from Theorem 1 follows that г d mu          2[m]

J_dS dV m ds x = ^ n I n ^{A u (0)} = ⁰ ’ m ^{— 3}

since 2 [ ^{m 1} = 0 at m — 3 (see example 3). In general case, if the function u ( x ) is a k -harmonic in S and u E C ^ ( S ) , them from Theorem 1 it follows that

Г d ^mu ,         k^{- 1} (2 i ) 1 ^m ] A i u (0)

JS dV m ^ds x = “ n g (2 - 2) , ( n- 2) , "° ^{- m — 2 k - 1}

because of equality (2 k — 2) [ ^{m 1} = 0 provided that m — 2 k — 1 .

2.    The value of polyharmonic function at the unit ball center

The following statement is true.

Theorem 2. For any polyharmonic in the unit ball S C R ⁿ function u E C ^{k- 1} ( S ) the equality

1 Г Л_{0 L1} du         k^_ ₁ d ^{k 1} u\

u(0) = “nL hhku + hk dv +'" + hk dV—1)dsx holds, where hsk are found from the equality

,     ( - 1) ^{k- 1} / 1 , _    \< k^{- 11}

h k    s !( k - 1)! g^(Xt ))„ =1 ^-

satisfy to the recurrence relation

h k +1 ^_ ( 1

-

S is 1 _{s -} 1 2 k )h ^{k +} 2 k h ^{k -}

and are coefficients of the polynomial

( - 1) ^{k 1}

H k- 1 ⁽ A ) = , _w⁽ A ^{— 2)} ••• ⁽ X ^- 2 к + 2)                        ⁽¹¹⁾

(2 к 2)..

expanded in the terms of factorial powers X ^{1 s} ]

H k- 1 ( X ) = h kk^{- 1} X 1 ^{k- 1} + h ^k _k ^{- 2} X 1 ^{k- 2} + • • • + h k X ^[1] + h k .                   (12)

The original proof of this Theorem is omitted because it requires some additional investigations and moreover this Theorem is a special case of more general Theorem 4.

It is necessary to note that recurrence relation similar to (10) a k ₊₁ = ( к — 2 s + 1) a k + 2 a k- ¹ was used in [7], where special polynomials were constructed. Regularization of integral equations was considered in [8, 9]. Recurrence relation of the form (10) determines some arithmetical triangle similar to Pascal, Euler and Stirling triangles, but its elements are rational fractions. Calculating h ^s _k by the formula (10) the triangle H can be written in the form

H =

11   3

16 16

93   29

^— 48

128 128    192 384

••• h k +1 = (1 — s/ (2 к )) h k — 1 / (2 к ) h k^{- 1} •••

Remark 1. Formula (8) according to (12) and (4) can be represented in the form u (0) =

^ω n ∂S

H k- 1 (Л) u ( x ) ds x .

Example 2. For a 4-harmonic function u E C ³ ( S ) , according to 4th row of the triangle H from (13), the following equality holds

3 d ² u   1 d ³ u

16 dv ²    48 dv ³

u (0) =

M u—— 11 du +

Ш п dss     16 dv

Consider polynomial

H ^m ) ( X ) = X ( X - 2) • • • ( X - 2 m + 2)( X - 2 m - 2) • • • ( X - 2 к + 2) .           (14)

It is obvious, that H k- 1 ( X ) = H k-^ X ) /H k- ₁(0) and H k ( m_ ) (2 m ) = 0 .

Lemma 1. Let u (x) = u 0 (x) + • • • + |x|2 k-2 uk-1( x)

be the Almansi representation of a к-harmonic in S function u(x) and such that u E Ck- 1(S), then for m E No and m < к the equality um(0) =

Ш n H km i (2 m )

/

∂S

H km ¹ (Л) u ( x ) ds x

holds true.

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ

Proof. Let A E R , i E N g , i < k and v ( x ) be a harmonic in S function. It is not hard to see that in S the equality

(Л — A ) ( | x | ² i v ( x ) ) = |x| ² ' ( (2 i — A ) v ( x ) + Л v ( x ) )

holds true and therefore

H-m1 (Л) (|x|2iv(x)) = |x|2' H-m1 (2i)v(x) + Qk- 1(Л)v(x)), where Q--1(A) is a certain polynomial of degree (k — 1) depending on H-m) and such that Q--1(0) = 0. Function Q-- 1(Л)v is also a harmonic in S function. Let Sr be a sphere of the radius r with a center at the origin of coordinates. For all r E (0, 1) we have Q--1 (Л)v E C(Sr).

Then

/

∂S r

Q -- 1 (Л) v ( x ) ds x

= Q -- 1 (0) [

∂S r

v ( x ) ds x = 0 .

Therefore, if i = m , then H -^{m )} (2 i ) = 0 and then

[ H -m ) (Л) f |x| ² i v ( x ) ) ds x = H -m ) (2 i )/ v ( x ) ds x ∂S r                                          ∂S r

+ j Q k- 1 (Л) v ( x ) ds x =0 .

If i = m then similarly to the above

4 [ H -m ¹ (Л) ( |x| ^{2 m} v ( x ) ) ds x = H -m ¹ (2 m )^ / ω n^r ∂S r                                       ω n^r ∂S

v ( x ) ds x = H -m 1 (2 m ) v (0) , r

where ш П is the surface area of the sphere dS r . Therefore for the function u ( x ) the equality

ω n ∂S r

H -m ¹ (Л) u ( x ) ds x

-- 1

= X .        H -m ¹ (Л) ( ix| ² ' “ ' ( x ))

i =0 ^ш п ^dSr_r          ^{V          7}

ds x = H -m ¹ (2 m ) U m (0) .

holds. Since u E C^{к 1}(- S ) , then dividing this equality on H -m ) (2 m ) = 0 and taking the limit as r ^ 1 we obtain the lemma’s statement (15).

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Theorem 3. For any k-harmonic in the unit ball S function u E C^k ( S ) the equality