Mean-Value Theorems for Harmonic Functions on the Cube in Rⁿ

Petar Petrov^*

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

DOI: 10.4236/apm.2015.511062                

Abstract

Let be a hypercube in Rⁿ. 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 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

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 Rⁿ. Section 3 deals with the best one-sided L¹-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 L¹-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 L¹-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 L¹-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 L¹-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 L¹-approximant from below to f with respect to.

Corollary 2 If, any solution h of the problem

(10)

is a best one-sided L¹-approximant from below to f with respect to.

Corollary 3 If, where and on, then is

a best one-sided L¹-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 L¹-

appro-ximant from below to f₁ 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 L¹-approximation to f₁ with the corresponding approximation from the linear sub- space of:

The possibility for explicit constructions of best one-sided L¹-approximants from, is studied in [10] . The functions and, where and are the unique best one-sided L¹-approximants to f₁ with respect to from below and above, respectively, play the role of basic error functions of the cano- nical one-sided L¹-approximation by elements of. For instance, can be constructed as the unique interpolant to f₁ on the boundary of the inscribed square and

(Figure 2).

Example 2 Let, and. The solution

of (10) with is a best one-sided L¹-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 L¹-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 L¹-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 L¹-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.

Conflicts of Interest

The authors declare no conflicts of interest.

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