**Advances in Pure Mathematics**

Vol.05 No.11(2015), Article ID:59617,5 pages

10.4236/apm.2015.511062

Mean-Value Theorems for Harmonic Functions on the Cube in

Petar Petrov

Faculty of Mathematics and Informatics, Sofia University, Sofia, Bulgaria

Email: peynov@fmi.uni-sofia.bg

Copyright © 2015 by author and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

Received 4 August 2015; accepted 13 September 2015; published 16 September 2015

ABSTRACT

Let
be a hypercube in. We prove theorems concerning mean-values of harmonic and polyharmonic functions on, which can be considered as natural analogues of the famous Gauss surface and volume mean-value formulas for harmonic functions on the ball in
and their extensions for polyharmonic functions. We also discuss an application of these formulas―the problem of best canonical one-sided L^{1}-approximation by harmonic functions on.

**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

and its boundary, based on integration over hyperplanar subsets of
and exact for harmonic or polyharmonic functions. They are presented in Section 2 and can be considered as natural analogues on
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 R^{n}. Section 3 deals with the best one-sided L^{1}-approximation by harmonic functions.

Let us remind that a real-valued function f is said to be harmonic ( polyharmonic of degree) in a given domain if and on, where is the Laplace operator and is its m-th iterate

For any set, denote by the linear space of all functions that are har- monic (polyharmonic of degree m) in a domain containing D. The notation will stand for the Lebesgue measure in.

2. Mean-Value Theorems

Let and be the ball and the hypersphere in

with center 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 both the average over and the average over are equal to.

The surface mean-value theorem. If, then

(1)

where is the -dimensional surface measure on the hypersphere.

The volume mean-value theorem. If, then

(2)

The balls are known to be the only sets in satisfying the surface or the volume mean-value theorem. This means that if is a nonvoid domain with a finite Lebesgue measure and if there exists a point

such that for every function h which is harmonic and integrable on, then is an

open ball centered at (see [4] ). The mean-value properties can also be reformulated in terms of quadrature domains [5] . Recall that is said to be a quadrature domain for, if is a connected open set

and there is a Borel measure with a compact support such that for every -

integrable harmonic function f on. Using the concept of quadrature domains, the volume mean-value property is equivalent to the statement that any open ball in
is a quadrature domain and the measure
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
is not a quadrature domain. Nevertheless, it is proved in Theorem 1 below that the closed hypercube
is a quadrature set in an extended sense, that is, we find explicitly a measure
with a compact support
having the above property with
replaced by
but the condition
is violated exactly at the “corners” (for the existence of quadrature sets see [7] ). This property of
is of crucial importance for the best one-sided L^{1}-approximation with respect to
(Section 3).

Let us denote by the -dimensional hyperplanar segments of defined by

(see Figure 1). Denote also

and. It can be calculated that

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

and

The following holds true.

Theorem 1 If, then h satisfies:

(i) Surface mean-value formula for the hypercube

(3)

(ii) Volume mean-value formula for the hypercube

(4)

In particular, both surface and volume mean values of h are attained on.

Proof. Set

and

Using the harmonicity of h, we get for

Hence, we have

(5)

if and

(6)

if.

Clearly, (5) is equivalent to (3) and from (6) it follows

(7)

which is equivalent to (4). □

Let. Analogously to the proof of Theorem 1 (ii), Equation (7) is generalized to:

Corollary 1 If and is such that and, then

(8)

The volume mean-value formula (2) was extended by P. Pizzetti to the following [2] [3] [8] .

The Pizzetti formula. If, then

Here, we present a similar formula for polyharmonic functions on the hypercube based on integration over the set.

Theorem 2 If, , and is such that, , then the following identity holds true for any:

(9)

where.

Proof. Equation (9) is a direct consequence from (8):

3. A Relation to Best One-Sided L^{1}-Approximation by Harmonic Functions

Theorem 1 suggests that for a certain positive cone in
the set
is a characteristic set for the best one-sided L^{1}-approximation with respect to
as it is explained and illustrated by the examples presented below.

For a given, let us introduce the following subset of:

A harmonic function
is said to be a best one-sided L^{1}-approximant from below to f with respect to
if

where

Theorem 1 (ii) readily implies the following ([6] [9] ).

Theorem 3 Let
and. Assume further that the set
belongs to the zero set of the function. Then
is a best one-sided L^{1}-approximant from below to f with respect to.

Corollary 2 If, any solution h of the problem

(10)

is a best one-sided L^{1}-approximant from below to f with respect to.

Corollary 3 If, where and on, then is

a best one-sided L^{1}-approximant from below to f with respect to.

Example 1 Let, and. By Corollary 2, the solution

of the interpolation problem (10) with
is a best one-sided L^{1}-

appro-ximant from below to f_{1} with respect to
and. Since the function
belongs

to the positive cone of the partial differential operator (that is,), one can compare

the best harmonic one-sided L^{1}-approximation to f_{1} with the corresponding approximation from the linear sub- space of:

The possibility for explicit constructions of best one-sided L^{1}-approximants from, is studied in [10] . The functions
and, where
and
are the unique best one-sided L^{1}-approximants to f_{1} with respect to
from below and above, respectively, play the role of basic error functions of the cano- nical one-sided L^{1}-approximation by elements of. For instance,
can be constructed as the unique interpolant to f_{1} on the boundary
of the inscribed square and

(Figure 2).

Example 2 Let, and. The solution

of (10) with
is a best one-sided L^{1}-approximant from

below to with respect to and. It can also be verified that (see Figure 3).

Figure 2. The graphs of the function
(coral) and its best one-sided L^{1}-approximants from below,
with respect to
(left) and
with respect to
(right).

Figure 3. The graphs of the function
(coral) and its best one-sided L^{1}-approximants from below,
with respect to
(left) and
with respect to
(right).

Remark 1 Let
is such that,
, and,
on. It follows from (8) that Theorem 3 also holds for the best weighted L^{1}-approximation from below with respect to
with weight. The smoothness requirements were used for brevity and wherever possible they can be weakened in a natural way.

Cite this paper

PetarPetrov, (2015) Mean-Value Theorems for Harmonic Functions on the Cube in R^{n}. *Advances in Pure Mathematics*,**05**,683-688. doi: 10.4236/apm.2015.511062

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