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MSC 31B30
ON THE MEAN-VALUE PROPERTY FOR POLYHARMONIC FUNCTIONS
V.V. Karachik
The mean-value property for normal derivatives of polyharmonic function on the unit sphere is obtained. The value of integral over the unit sphere of normal derivative of mth order of polyharmonic function is expressed through the values of the Laplacian’s powers of this function at the origin. In particular, it is established that the integral over the unit sphere of normal derivative of degree not less then 2k — 1 of k-harmonic function is equal to zero. The values of polyharmonic function and its Laplacian’s powers at the center of the unit ball are found. These values are expressed through the integral over the unit sphere of a linear combination of the normal derivatives up to k — 1 degree for the k-harmonic function. Some illustrative examples are given.
Keywords: polyharmonic functions, mean-value property, normal derivatives on a sphere.
Introduction
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 M™ and Br(x) = {y E M™ : \y — x\ < r}. It is well known (see [1]) the Gauss mean-value property for harmonic functions: if x E Q and Br(x) C Q, then for all functions harmonic in Q
u(x) = dR(x)\ u(y) dsy ■ (1)
\dBr(x)\ JdBr(x)
This mean-value property has been extended by Pizzetti (see [2]) for k-harmonic functions in Q to the form
k-l
____1____[ u(y)ds =r(n/2)T r2iAtu(xo)
\dBr (x)\ JdBr(x) y (/) i=0 4^(1 + n/2))
where r(a) is the Euler’s gamma function. This property can be easily written for a k-harmonic function u E Ck-1(S) in the unit ball S C M™ in the form
1 f . . , k— Aiu(0)
— u(x) dsx = > ,——, (2)
^™ JdS (2, 2)i(n, 2)i
where u™ is the surface area of the unit sphere dS, and (a, b)k = a(a + b) ■ ■ ■ (a + b(k — 1)) is the generalized Pochhammer symbol with (a,b)o = 1. For example, (2, 2)i = (2i)W. The similar formula was proved in [3, Theorem 7] for calculating the integral of homogeneous polynomial Qm(x) on the unit sphere
^ . 0, m E 2N — 1
/ Qm(x) dsx = \ Am/2Qm(x) 2^
j\x\=i I —T.---------7—■-----T7 u™, m E 2N
1 m!! n ■ ■ ■ (n + m — 2)
Consider the operator A defined by the equality
(3)
This operator plays an important role in our investigation because in the paper [4] it was proved that the following equality is fulfilled on dS
Besides, it is known (see, for example, [5]) that if u is a harmonic function, then function P(A)u is also a harmonic one, where P(X) 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 No = N U {0}.
Theorem 1. For all m E No and for any polyharmonic function in the unit ball u E Cm(S) the following equality holds
where v is the unit outer normal to dS.
Proof. In [6, Theorem 4] it is proved that for any polyharmonic in S function u(x) the following Almansi representation takes place
(4)
where v is the outer normal to dS, and t[k = t(t — 1)... (t — k + 1) is a factorial power of t.
(6)
where harmonic in S functions vo(x),..., Vk(x),... are given by the formula
(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 ko such that Vk(x) = 0 for all k > ko. It is not hard to see that
A(\x\2k u) = \x\2k (2k + Л)и
and therefore
Л[2 (\x\2k u) = (Л — 1) (\x\2k (2k + Л)и) = \x\2k (2k — 1 + Л)(2к + Л)и,
whence
ЛН (\x\2ku) = \x\2k(2k — m + 1 + Л) • • • (2k — 1 + Л)(2к + Л)и = (2k)[m]u + Ят(Л)и,
where Qm(X) is a certain polynomial such that Qm(0) = 0. Therefore in S we have
лНи^) = ЛН^) + ^ (2k4:k'xP 0 (1(k -1),1 an/2-1Vk(ax) da+
k=1
^ i 12k r 1/i „,\k- 1
^ |x|2k r1 (1 _ a)k-1
+ 2 4kk! I k — 1)! an/2-lQm(A)vk(ax) da.
Using the mean-value property for harmonic functions and harmonicity of Avk we can obtain
the equality f9s Qm(A)vk(ax) dsx = 0. Therefore using (7) we have
— f A[m1u(x) dsx = £ /1(1 — a)k-1an/2-1 davk(0) =
Wn JdS k=1 4kk!(k — 1)! 'o
OO
= (2k)[m1vk(0) r(k)r(n/2) = ^ (2k)[m1vk(0) = ^ (2k)m1Aku(0)
= 2=1 4kk!(k — 1)! r(k + n/2) = (2, 2)k(n, 2)k = k= (2,2)k(n, 2)k .
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.
□
Example 1. Let function u(x) be a harmonic in S and u E C^(S), then from Theorem 1 follows that
r dmu ,
/ t;—dsx = 0, m > 1.
JdS dvm -
For a biharmonic in S function u E C^(S) from Theorem 1 follows that
f d mu 2m1
dsx = Au(0) = 0, m > 3
Jqs dvm 2n
since 2[m1 =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
д mu — (2г)Нд* и(0)
ids m
m
и , V—> I2ir ;Д ЩО)
dsx = шп } —---------- ------— = 0, m > 2k — 1
x n (2 2),(n 2) ’ >
dvm x n^n (2,2)i(n, 2)
i=o
because of equality (2k — 2)[m1 = 0 provided that m > 2k — 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 Rn function u E Ck-1(S) the equality
1 f / du dk 1u \
u(0) = wn L (hku + hkdv + • • ■ + hk-1 dVk-) dsx (8)
holds, where hk are found from the equality
(_1)k-1 /1 \ (k-1)
hk=• (9)
satisfy to the recurrence relation
h' . , = (1 - , ,
2kJ k 2k
hk+1 = f1 — hk + 27;hk 1, (10)
and are coefficients of the polynomial
( —1)k-1
Hk-1(X) = (2k — 2)!!(X — 2) ■ ■■(X — 2k + 2) (11) expanded in the terms of factorial powers Xs1
Hk-1(X) = hk-1X[k-1] + hkk-2X[k-2] + ■ ■ ■ + h\X[11 + h°°. (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) ak+1 = (k—2s + 1)ak +1 ak-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 hk by the formula (10) the triangle H can be written in the form
1
1
1------
! —828
8 8
H = 1 A (13)
16 16 48
93 29 7 1
128 128 192 384
1
■ ■ ■ hk+1 = (1 — s/(2k))hk — 1/(2k)hk~
Remark 1. Formula (8) according to (12) and (4) can be represented in the form
и(0) = — I Hк-1(Л)u(x) dsx
шп JdS
1
Wn
Example 2. For a 4-harmonic function u E C3(S), according to 4th row of the triangle H from (13), the following equality holds
u(0) = f ( un JdS V
1 f ( 11 ди 3 д2и 1 д3и \ d
шп ds V 16 ди + 16 ди2 48 ди3/ Sx
Consider polynomial
Hkm1 (X) = X(X — 2) ■ ■ ■ (X — 2m + 2)(X — 2m — 2) ■ ■ ■ (X — 2k + 2). (14)
It is obvious, that Hk-1(X) = Hk-)1(X)/Hk-)1(0) and Hjk--1 (2m) = 0.
Lemma 1. Let
u(x) = uo (x) + ■ ■ ■ + |x|2k-2uk-1(x)
be the Almansi representation of a k-harmonic in S function u(x) and such that u E Ck-1(S), then for m E No and m < k the equality
um(0) = ------7^--I Hkm1(A)u(x) dsx (15)
WnH- (2m) JdS
holds true.
Proof. Let X E R, i E No, i < k and v(x) be a harmonic in S function. It is not hard to see that
in S the equality / \
(A — X) ^ |x|2iv(x)j = |x|2i((2i — X)v(x) + Av(x))
holds true and therefore
H^ (A)( |x|2iv(x)) = |x|2i( H^ (2i)v(x) + Qk-1(A)v(x^,
where Qk-1(X) is a certain polynomial of degree (k — 1) depending on Hkm) and such that Qk-1(0) = 0. Function Qk-1(A)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 Qk-1(A)v E C(Sr). Then
/ Qk-1(A)v(x) dsx = Qk-1(0) v(x) dsx = 0.
JaSr JdSr
Therefore, if i = m, then Hkm)(2i) = 0 and then
[ H^(A)f|x|2iv(x)) dsx = Hm\(2i) / v(x) dsx + / Q-(A)v(x) dsx = 0.
JdSr V ' JdSr JdSr
If i = m then similarly to the above
-7 / Hkm)(a) (|x|2™v(x^ dsx=Ht-1(2m)-7 f v(x) dsx=Hk-1(2m)v(0),
Wn JdSr V ' Wn JdSr
where w7n is the surface area of the sphere dSr. Therefore for the function u(x) the equality
k— 1
^7 f Hkm)(A)u(x) dsx = ^ f Hk-1(A) (|x|2iui(x^ dsx = Hkm)(2m)um(0). (16)
Wn JdSr Wn JdSr V '
holds. Since u E Ck 1(»S), then dividing this equality on Hkm) (2m) = 0 and taking the limit as r -— 1 we obtain the lemma’s statement (15).
□
Theorem 3. For any k-harmonic in the unit ball S function u E Ck(S) the equality
f Hkk)^)u(x) dsx = 0, JdS
holds, where Hkf)(X) = X(X — 2) ■ ■ ■ (X — 2k + 2).
Proof. It is not hard to see that V i < m, H^ (2i) = 0. Therefore using the equality (16) from Lemma 1 at r E (0, 1) we have
k— 1
f Hkk)(A)u(x) dsx = £ f Hkk) (A) (|x|2iui(x)) dsx = 0.
Wn J dSr i_o Wn J dSr
Taking the limit for r - 1 we obtain the desired equality.
□
Some generalization of the well known property of the harmonic functions Jds dU dsx =0 on the polyharmonic functions is the following assertion.
Sequence 1. If the numbers ai are found from the equality
(k)
Hkk)(X) = A[k] + ak-1\[k-1] + • • • + a^1 +
ao,
then for any polyharmonic in the unit ball S function u E Ck (S) the equality
d k u'
holds.
To prove this corollary it is sufficient to remember (4) and to take advantage of Theorem 3. Theorem 2 can be generalized in the following way.
Theorem 4. For any polyharmonic in the unit ball S function u E Ck-1(S) the equality
Amu(0) = -(2~ 2(),;;i("-2)m / h- (amx (17)
Wn H- (2m) Jds holds, where the polynomial Hkm) (X) is defined in (14) and m = 0,... ,k — 1.
Proof. Let
u(x) = uo (x) + ■ ■ ■ + |x|2k-2uk-1(x) (18)
be the Almansi representation of a k-harmonic in S function u(x) and such that u E Ck-1(S),
then for m E No and m < k the equality (15) holds. Besides, if v is a harmonic in S function,
then (see [4])
A^ |x|2mv(x) j = |x|2m-22m(2m + n — 2 + 2A)v(x).
Therefore for i < m we have
,
2, 2i
A^ |x|2mv(x)) = |x|2m-2i \[ 2j(2j + n — 2 + 2A)v(x)
j_m-i+1
and hence Am Ixl2mv(x)) = 0. If i = m, then we have
|x_o
m m
Am^|x|2mv(x^ = 2j(2j + n — 2 + 2A)v(x) = 2j(2j + n — 2)v(x) + Pk(A)v(x) =
j_1 j_1
= 2m!! n ■■■ (n + 2m — 2)v(x) + Pk (A)v(x) = (2,2),(n, 2),v(x) + Pk (A)v(x), (19)
where Pk(X) is a certain polynomial of kth power and such that Pk(0) = 0. Therefore we obtain
Am(|x|2mv(x^ = (2, 2);(n, 2),v(0).
|x_o
From (18) it follows that for i > m
A^ |x|2m v(x)) = (2,2)m(n, 2)mAi-mv(x) + Ai-mPk (A)v(x) = 0.
Therefore applying the operator Am for m E No and m < k to the equality (18), then assuming x = 0 and using (15) we obtain
Amu(x)|„o = £Am(|x|2iMx)) o = —<2’2h(n'2)m f H,(A)u(x)ds,.
i_o v /|x_o un Hkmi (2m) Jas
Formula (17) is proved.
□
Remark 2. It is not hard to see that if the function u E Ck(S) is a (k + 1)-harmonic in the unit ball, then for numbers ai from Sequence 1, according to Theorem 4, the following equality holds
d k u\
dsx.
f / ди дk u \
Ak u(0) = Jd yaou + a1 — +--------------------+ ak J
Example 3. Let the function u(x) be a 3-harmonic one in S and u E C2(S). It is easy to see that
h22)(X) = X(X — 2) = —X[1] + X[2],
(2)
H2 (4) = 8, (2, 2)2 = 8, (n, 2)2 = n(n + 2) and therefore the following equality holds
. 2 . . n(n + 2) f ( du d2u \ ,
A2u(0) = —-----------!- — — + — dsx.
Un Jasv dv dv2)
If the function u(x) is a biharmonic one in S, then according to Sequence 1 we obtain
f ( du d2u ^ f du f d2u
— TT + Tj o dsx = 0 ^ — dsx = / Tj o dsx.
Jqs\ dv dv2) Jos dv Jos dv2
References
1. Stein E.M., Weiss G. Introduction to Fourier Analysis on Euclidian Spaces. Princeton Univ. Press, Princeton, NJ, 1971.
2. Dalmasso R. On the Mean-Value Property of Polyharmonic Functions. Studia Sci. Math. Hungar., 2010, vol. 47, no. 1, pp. 113-117.
3. Karachik V.V. On Some Special Polynomials and Functions. Siberian Electronic Mathematical Reports, 2013, vol. 10, pp. 205-226.
4. Karachik V.V. Construction of Polynomial Solutions to Some Boundary Value Problems for Poisson’s Equation. Computational Mathematics and Mathematical Physics, 2011, vol. 51, no. 9, pp. 1567-1587.
5. Karachik V.V. A Problem for the Polyharmonic Equation in the Sphere. Siberian Mathematical J., 2005, vol. 32, no. 5, pp. 767-774.
6. Karachik V.V. On One Representation of Analytic Functions by Harmonic Functions. Siberian Advances in Mathematics, 2008, vol. 18, no. 2, pp. 103-117.
7. Karachik V.V. On Some Special Polynomials. Proceedings of American Mathematical Society, 2004, vol. 132, no. 4, pp. 1049-1058.
8. MeHHxec ^.^. For Regularization of Certain Classes of Mappings Inverse to Integral Operators [O regulyarizuemosti nekotorykh klassov otobrazheniy, obratnykh k integral’nym operatoram]. Matematicheskie Zametki [Mathematical Notes], 1999, vol. 65, no. 2, pp. 222-229.
9. Menikhes L.D. On Sufficient Condition for Regularizability of Linear Inverse Problems. Mathematical Notes, 2007, vol. 82, no. 1-2, pp. 242-246.
УДК 517.575
О СВОЙСТВЕ СРЕДНЕГО ДЛЯ ПОЛИГАРМОНИЧЕСКИХ ФУНКЦИЙ В ШАРЕ
В.В. Карачик
Получено свойство среднего для нормальных производных от полигармонической функции по единичной сфере. Значение интеграла от нормальных производных по единичной сфере от полигармонической функции выражается через значения степеней лапласианов от этой функции в начале координат. В частности, установлено, что интеграл по единичной сфере от нормальных производных k-гармонической функции порядка не меньше 2k — 1 равен нулю. Найдены значения полигармонической функции и лапласианов от нее в центре единичного шара. Это значение выражается через интеграл по единичной сфере от линейной комбинации нормальных производных до k — 1 порядка для k-гармонической функции. Приведены иллюстративные примеры.
Ключевые слова: полигармонические функции, свойство среднего, нормальные производные на сфере.
Литература
1. Stein, E.M. Introduction to Fourier Analysis on Euclidian Spaces / E. M. Stein, G. Weiss.
- Princeton Univ. Press, Princeton, NJ, 1971.
2. Dalmasso, R. On the Mean-Value Property of Polyharmonic Functions / Dalmasso R. // Studia Sci. Math. Hungar. - 2010. - V. 47, №1. - P. 113-117.
3. Карачик, В.В. О некоторых специальных полиномах и функциях / В.В. Карачик // Сибирские электронные математические известия. - 2013. - Т. 10. - C. 205-226.
4. Карачик, В.В. Построение полиномиальных решений некоторых краевых задач для уравнения Пуассона / В.В. Карачик // ЖВМиМФ. - 2011. - Т. 51, №9. - C. 1674-1694.
5. Карачик, В.В. Об одной задаче для полигармонического уравнения в шаре / В.В. Карачик // Сибирский математический журнал, 1991. - Т. 32, №5. - С. 51-58.
6. Карачик, В.В. Об одном представлении аналитических функций гармоническими / В.В. Карачик // Математические труды. - 2007. - Т. 10, №2. - C. 142-162.
7. Karachik, V.V. On Some Special Polynomials / V.V. Karachik // Proceedings of American Mathematical Society. - 2004. - V. 132, №4. - P. 1049-1058.
8. Менихес, Л.Д. О регуляризуемости некоторых классов отображений, обратных к интегральным операторам / Л.Д. Менихес // Математические заметки. - 1999. - Т. 65, №2.
- С. 222-229.
9. Menikhes, L.D. On Sufficient Condition for Regularizability of Linear Inverse Problems / L.D. Menikhes // Mathematical Notes. - 2007. - V. 82, №1-2. - P. 242-246.
Валерий Валентинович Карачик, доктор физико-математических наук, профессор, кафедра математического анализа, Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), karachik@susu.ru.
Поступила в редакцию 29 апреля 2013 г.
