Mean-Value Theorems for Harmonic Functions on the ...

Advances in Pure Mathematics, 2015, 5, 683-688 Published Online September 2015 in SciRes. http://www.scirp.org/journal/apm http://dx.doi.org/10.4236/apm.2015.511062

Mean-Value Theorems for Harmonic Functions on the Cube in  n Petar Petrov Faculty of Mathematics and Informatics, Sofia University, Sofia, Bulgaria Email: [email protected] Received 4 August 2015; accepted 13 September 2015; published 16 September 2015 Copyright © 2015 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Abstract

{

}

Let I n ( r ) = x ∈  n xi ≤ r , i =1, 2, , n

be a hypercube in  n . We prove theorems concerning

mean-values of harmonic and polyharmonic functions on I n ( r ) , which can be considered as natural analogues of the famous Gauss surface and volume mean-value formulas for harmonic functions on the ball in  n and their extensions for polyharmonic functions. We also discuss an application of these formulas—the problem of best canonical one-sided L1-approximation by harmonic functions on I n ( r ) .

Keywords Harmonic Functions, Polyharmonic Functions, Hypercube, Quadrature Domain, Best One-Sided Approximation

1. Introduction This note is devoted to formulas for calculation of integrals over the n-dimensional hypercube centered at 0

{

}

I n := I n ( r ) := x ∈  n xi ≤ r , i =1, 2, , n , r > 0, and its boundary Pn := Pn ( r ) := ∂I n ( r ) , based on integration over hyperplanar subsets of I n and exact for harmonic or polyharmonic functions. They are presented in Section 2 and can be considered as natural analogues on I n of Gauss surface and volume mean-value formulas for harmonic functions ([1]) and Pizzetti formula [2], ([3], Part IV, Ch. 3, pp. 287-288) for polyharmonic functions on the ball in n. Section 3 deals with the best one-sided L1-approximation by harmonic functions. n

How to cite this paper: Petrov, P. (2015) Mean-Value Theorems for Harmonic Functions on the Cube in  . Advances in Pure Mathematics, 5, 683-688. http://dx.doi.org/10.4236/apm.2015.511062

P. Petrov

Let us remind that a real-valued function f is said to be harmonic ( polyharmonic of degree m ≥ 2 ) in a given 0 on Ω , where ∆ is the Laplace domain Ω ⊂  n if f ∈ C 2 ( Ω ) f ∈ C 2m ( Ω ) and ∆f = 0 ∆m f = m operator and ∆ is its m-th iterate

(

)

(

)

∂2 f , ∆ m f :=∆ ∆ m −1 f . 2 ∂ x i =1 i

(

n

∆f :=∑

(

)

)

For any set D ⊂  n , denote by H ( D ) H m ( D ) , m ≥ 2 the linear space of all functions that are harmonic (polyharmonic of degree m) in a domain containing D. The notation dλn will stand for the Lebesgue measure in  n .

2. Mean-Value Theorems

{

(

x ∈  n x := Let Bn ( r ) := ∑ i =1xi2 n

)

12

≤r

}

r} be the ball and the hypersphere in and Sn ( r ) := {x ∈ n x =

 n with center 0 and radius r. The following famous formulas are basic tools in harmonic function theory and state that for any function h which is harmonic on Bn ( r ) both the average over Sn ( r ) and the average over Bn ( r ) are equal to h ( 0 ) . The surface mean-value theorem. If h ∈ H ( Bn ( r ) ) , then 1

hdσ n −1 = h ( 0 ) ,

σ n −1 ( Sn ( r ) ) ∫Sn ( r )

(1)

where dσ n −1 is the ( n − 1) -dimensional surface measure on the hypersphere Sn ( r ) . The volume mean-value theorem. If h ∈ H ( Bn ( r ) ) , then

1

hdλn = h ( 0 ) .

λn ( Bn ( r ) ) ∫Bn ( r )

(2)

The balls are known to be the only sets in  n satisfying the surface or the volume mean-value theorem. This means that if Ω ⊂  n is a nonvoid domain with a finite Lebesgue measure and if there exists a point x0 ∈ Ω 1 h dλn for every function h which is harmonic and integrable on Ω , then Ω is an such that h ( x0 ) = λn ( Ω ) ∫Ω open ball centered at x0 (see [4]). The mean-value properties can also be reformulated in terms of quadrature domains [5]. Recall that Ω ⊂  n is said to be a quadrature domain for H ( Ω ) , if Ω is a connected open set and there is a Borel measure dµ with a compact support K µ ⊂ Ω such that ∫Ω f dλn = ∫K f dµ for every λn µ

integrable harmonic function f on Ω . Using the concept of quadrature domains, the volume mean-value property is equivalent to the statement that any open ball in  n is a quadrature domain and the measure dµ is the Dirac measure supported at its center. On the other hand, no domains having “corners” are quadrature domains [6]. From this point of view, the open hypercube I n is not a quadrature domain. Nevertheless, it is proved in Theorem 1 below that the closed hypercube I n is a quadrature set in an extended sense, that is, we find explicitly a measure dµ with a compact support K µ having the above property with Ω replaced by I n but the condition K µ ⊂ I n is violated exactly at the “corners” (for the existence of quadrature sets see [7]). This property of I n is of crucial importance for the best one-sided L1-approximation with respect to H ( I n ) (Section 3). Let us denote by Dnij the ( n − 1) -dimensional hyperplanar segments of I n defined by

Dnij := Dnij ( r ) :=

{x ∈ I

n

}

xk ≤ xi = x j , k ≠ i, j , 1 ≤ i < j ≤ n,

(see Figure 1). Denote also = ωk ( x ) :

( r − max { x

1

, x2 , , xn } k!

and dλmωk := ωk dλm . It can be calculated that

684

)

k

, k ≥ 0,

P. Petrov

Figure 1. The sets D312 (1) (white), D313 (1) (green) and D323 (1) (coral).

r n+k r n + k −1 n , λnω−k1 ( Pn ) 2n n ! , n! = λnωk ( I n ) 2= ( n + k )! ( n + k − 1)! and

r n + k −1 n −1 n! = λnω−k1 ( Dn ) 2= , where Dn : ( n + k − 1)!

1≤i < j ≤ n Dnij .

The following holds true. Theorem 1 If h ∈ H ( I n ) , then h satisfies: (i) Surface mean-value formula for the hypercube 1 1 hdλn −1 = hdλn −1 , ∫ P n λn −1 ( Pn ) λn −1 ( Dn ) ∫Dn (ii) Volume mean-value formula for the hypercube 1 1 = h dλnωk h dλnω−k1+1 , k ≥ 0. ∫ ∫ ωk ωk +1 In Dn λn ( I n ) λn −1 ( Dn ) In particular, both surface and volume mean values of h are attained on Dn . Proof. Set = Mi : M = i ( x ) : max x j , j ≠i

and

xti := ( x1 , , xi −1 , t , xi +1 , , xn ) .

Using the harmonicity of h, we get for k ≥ 1

∂2h dλn n ∂xi2 i =1 n n r r ∂ω ∂h = −∑ ∫− r  ∫− r k ( x ) ( x ) dxi dx1  dxi −1dxi +1  dxn ∂xi ∂xi i =1 n r  r  r −Mi ∂h = −∑ ∫− r  ∫− r  ∫− r + ∫M sign xiωk −1 ( x ) ( x ) dxi  dx1  dxi −1dxi +1  dxn i x ∂ i =1 i   n

ω 0= ∑ ∫I ωk ∫I ∆hdλn k =

(

)

n r  r  r ∂  h x−i xi + h ( x )  dxi  dx1  dxi −1dxi +1  dxn . = −∑ ∫− r  ∫− r  ∫M ωk −1 ( x )   ∂xi i =1  i 

(

685

)

(3)

(4)

P. Petrov

Hence, we have n

{ ( ) ( ) ( ) ( )}

0= −∑ ∫− r  ∫− r h x−i r + h x+i r −  h x−i M i + h x+i M i  dx1  dxi −1dxi +1  dxn   i =1 r

r

(5)

if k = 1 and n

)

(

−∑ ∫  ∫ ∫ ωk − 2 ( x )  h x−i xi + h ( x )  dxi dx1  dxi −1dxi +1  dxn 0= −r −r Mi   i =1 r

n

r

r

(

) (

)

) (

(6)

+ ∑ ∫  ∫ ωk −1 x+i M i  h x−i M i + h x+i M i  dx1  dxi −1dxi +1  dxn −r −r   i =1 r

r

if k ≥ 2 . Clearly, (5) is equivalent to (3) and from (6) it follows ω ω ω 0= ∫I ∆hdλn k = ∫I hdλn k −2 − 2∫D hdλn −k1−1 , n

n

(7)

n

which is equivalent to (4). □ = M: M = Let ( x ) : max1≤i ≤ n xi . Analogously to the proof of Theorem 1 (ii), Equation (7) is generalized to: Corollary 1 If h ∈ H ( I n ) and ϕ ∈ C 2 [ 0, r ] is such that ϕ ( 0 ) = 0 and ϕ ′ ( 0 ) = 0 , then

0=

∫I ϕ ( r − M ) ∆hdλn= ∫I ϕ ′′ ( r − M ) hdλn − 2∫D ϕ ′ ( r − M ) hdλn −1. n

n

(8)

n

The volume mean-value formula (2) was extended by P. Pizzetti to the following [2] [3] [8]. The Pizzetti formula. If g ∈ H m ( Bn ( r ) ) , then

∆k g ( 0) r 2k . 2k k! k = 0 2 Γ ( n 2 + k + 1)

m −1

gdλn = r n π n 2 ∑ r

∫B ( ) n

Here, we present a similar formula for polyharmonic functions on the hypercube based on integration over the set Dn . k Theorem 2 If g ∈ H m ( I n ) , m ≥ 1 , and ϕ ∈ C 2 m [ 0, r ] is such that ϕ ( ) ( 0= ) = 0 , k 0,1, , 2m − 1 , then the following identity holds true for any k ≥ 0 :

∫I ϕ

(2m)

n

m −1

2∑ ∫ ϕ (

M ) gdλn ( r −=

s =0

2 s +1)

Dn

( r − M ) ∆ m − s −1 g dλn −1 ,

(9)

d jϕ (t ) . dt j Proof. Equation (9) is a direct consequence from (8):

where ϕ (

j)

(t ) =

∫I ϕ ( r − M ) ∆

0=

m

n

g dλn

= −2 ∫ nϕ ( ) ( r − M ) ∆ m −1 g dλn −1 + ∫ ϕ ( D I 1

2)

n

m −1

2 s +1)

= = −2 ∑ ∫ ϕ ( D s =0

n

( r − M ) ∆ m −1 g dλn

∆ m − s −1 g dλn −1 + ∫ ϕ ( In

2m)

g dλn .

3. A Relation to Best One-Sided L1-Approximation by Harmonic Functions Theorem 1 suggests that for a certain positive cone in C ( I n ) the set Dn is a characteristic set for the best one-sided L1-approximation with respect to H ( I n ) as it is explained and illustrated by the examples presented below. For a given f ∈ C ( I n ) , let us introduce the following subset of H ( I n ) :

{

}

h ∈ H ( I n ) h ≤ f on I n . H − ( I n , f ) := A harmonic function h*f ∈ H − ( I n , f ) is said to be a best one-sided L1-approximant from below to f with respect to H ( I n ) if

686

f − h*f

1

≤ f − h 1 for every h ∈ H − ( I n , f ) ,

P. Petrov

where

g 1 := ∫ g dλn . In

Theorem 1 (ii) readily implies the following ([6] [9]). Theorem 3 Let f ∈ C ( I n ) and h*f ∈ H − ( I n , f ) . Assume further that the set Dn belongs to the zero set of the function f − h*f . Then h*f is a best one-sided L1-approximant from below to f with respect to H ( I n ) . Corollary 2 If f ∈ C1 ( I n ) , any solution h of the problem

h| Dn = f | Dn , ∇h| Dn = ∇f | Dn , h ∈ H − ( I n , f ) ,

(10)

is a best one-sided L1-approximant from below to f with respect to H ( I n ) .

Corollary 3 If f ( x ) g ( x ) ∏1≤i < j ≤ n ( xi2 − x 2j ) , where g ∈ C ( I n ) and g ≥ 0 on I n , then h*f ( x ) ≡ 0 is = 2

a best one-sided L1-approximant from below to f with respect to H ( I n ) . Example 1 Let n = 2 , r = 1 and f1 ( x1 , x2 ) = x12 x22 . By Corollary 2, the solution 3 h*f1 ( x1 , x2 ) = − x14 4 + x12 x22 − x24 4 of the interpolation problem (10) with f = f1 is a best one-sided L12 8 45 . Since the function f1 belongs appro-ximant from below to f1 with respect to H ( I 2 ) and f1 − h*f1 = 1 4 ∂ 4 4 := 2 2 (that is, 2,2 to the positive cone of the partial differential operator 2,2 f1 > 0 ), one can compare ∂x1 ∂x2 the best harmonic one-sided L1-approximation to f1 with the corresponding approximation from the linear subspace of C ( I 2 ) : 1   B 2,2 ( I 2 ) := b ∈ C ( I 2 ) b ( x1 , x2 ) = ∑  a0 j ( x1 ) x2j + a1 j ( x2 ) x1j   . j =0   The possibility for explicit constructions of best one-sided L1-approximants from B 2,2 ( I 2 ) , is studied in [10]. The functions f1 − b*f1 and f1 − b*f1 , where b*f1 and b*f1 are the unique best one-sided L1-approximants to f1 with respect to B 2,2 ( I 2 ) from below and above, respectively, play the role of basic error functions of the canonical one-sided L1-approximation by elements of B 2,2 ( I 2 ) . For instance, b*f1 can be constructed as the unique = ◊ : ( x1 , x2 ) ∈ I 2 x1= + x2 1 of the inscribed square and interpolant to f1 on the boundary 14 45 (Figure 2). f1 − b*f1 = 1 Example 2 Let n = 2 , r = 1 and f 2 ( x1 , x2 ) = x18 + 14 x14 x24 + x28 . The solution h*f2 ( x1 , x2 ) = x18 + x28 − 28 x16 x22 + x12 x26 + 70 x14 x24 of (10) with f = f 2 is a best one-sided L1-approximant from

{

(

}

)

below to f 2 with respect to H ( I 2 ) and (see Figure 3).

f 2 − h*f2 = 8 75 . It can also be verified that

f 2 − b*f2 = 121 900

Figure 2. The graphs of the function f1 ( x1 , x2 ) = x12 x22 (coral) and its best one-sided L1-approximants from below, h*f1 with respect to H ( I 2 ) (left) and b*f1 with respect to B 2,2 ( I 2 ) (right).

687

P. Petrov

x18 + 14 x14 x24 + x28 (coral) and its best one-sided L1-approximants from Figure 3. The graphs of the function f 2 ( x1 , x2 ) = below, h*f2 with respect to H ( I 2 ) (left) and b*f2 with respect to B 2,2 ( I 2 ) (right).

Remark 1 Let ϕ ∈ C 2 [ 0, r ] is such that ϕ ( 0 ) = 0 , ϕ ′ ( 0 ) = 0 , and ϕ ′ ≥ 0 , ϕ ′′ ≥ 0 on [ 0, r ] . It follows from (8) that Theorem 3 also holds for the best weighted L1-approximation from below with respect to H ( I n ) with weight ϕ ′′ ( r − M ) . The smoothness requirements were used for brevity and wherever possible they can be weakened in a natural way.

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Helms, L.-L. (2009) Potential Theory. Springer-Verlag, London. http://dx.doi.org/10.1007/978-1-84882-319-8

[2]

Pizzetti, P. (1909) Sulla media dei valori che una funzione dei punti dello spazio assume sulla superficie della sfera. Rendiconti Linzei—Matematica e Applicazioni, 18, 182-185.

[3]

Courant, R. and Hilbert, D. (1989) Methods of Mathematical Physics Vol. II. Partial Differential Equations Reprint of the 1962 Original. John Wiley & Sons Inc., New York.

[4]

Goldstein, M., Haussmann, W. and Rogge, L. (1988) On the Mean Value Property of Harmonic Functions and Best Harmonic L1-Approximation. Transactions of the American Mathematical Society, 305, 505-515.

[5]

Sakai, M. (1982) Quadrature Domains. Lecture Notes in Mathematics, Springer, Berlin.

[6]

Gustafsson, B., Sakai, M. and Shapiro, H.S. (1977) On Domains in Which Harmonic Functions Satisfy Generalized Mean Value Properties. Potential Analysis, 71, 467-484.

[7]

Gustafsson, B. (1998) On Mother Bodies of Convex Polyhedra. SIAM Journal on Mathematical Analysis, 29, 11061117. http://dx.doi.org/10.1137/S0036141097317918

[8]

Bojanov, B. (2001) An Extension of the Pizzetti Formula for Polyharmonic Functions. Acta Mathematica Hungarica, 91, 99-113. http://dx.doi.org/10.1023/A:1010687011674

[9]

Armitage, D.H. and Gardiner, S.J. (1999) Best One-Sided L1-Approximation by Harmonic and Subharmonic Functions. In: Haußmann, W., Jetter, K. and Reimer, M., Eds., Advances in Multivariate Approximation, Mathematical Research (Volume 107), Wiley-VCH, Berlin, 43-56.

[10] Dryanov, D. and Petrov, P. (2002) Best One-Sided L1-Approximation by Blending Functions of Order (2,2). Journal of Approximation Theory, 115, 72-99. http://dx.doi.org/10.1006/jath.2001.3652

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