# Green function of the Laplacian for the Neumann problem in [R.sup.n.sub.+].

1 Introduction

Let [OMEGA] be a domain of [R.sup.n] with smooth boundary [partial derivative][OMEGA].

Recall the Dirichlet and Neumann problems for the Laplacian.

(1) Dirichlet Problem (DP):

Find u [member of] [C.sup.2]([OMEGA]) [intersection] C([OMEGA])) such that

[DELTA]u = f in [OMEGA], u = g on [partial derivative][OMEGA]. (1)

(2) Neumann Problem (NP):

Find u [member of] [C.sup.2]([OMEGA]) [intersection] [C.sup.1]([bar.[OMEGA]]) such that

[DELTA]u = f in [OMEGA] = f in [OMEGA], du/d[[eta].sub.x] = [g.sub.1] on [partial derivative][OMEGA], (2)

where [[eta].sub.x] is the outward unit normal to [partial derivative][OMEGA] at x [member of] [partial derivative][OMEGA], du/d[[eta].sub.x] is the normal derivative of u at x, f : [OMEGA] [right arrow] R, and g, [g.sub.1] : [partial derivative][OMEGA] [right arrow] R are prescribed continuous functions.

By analogy with the Green function for the Dirichlet problem for a domain [OMEGA], we consider the Green type function for the Neumann problem for [OMEGA] (also known as Neumann's function, or Green's function for the Neumann problem or Green's function of the second kind). We will give the explicit forms of the Neumann's functions and the solution of the Neumann problem for the upper half-space of the n-th dimensional euclidian space [R.sup.n], n [greater than or equal to] 2.

The concept of Green type function for (NP) has been considered by several authors (, , , , , , , , , ).

In , there are presented the expressions of the Green's functions for the Neumann's problem for a ball in [R.sup.2] and [R.sup.3]. In , Neumann's function for the sphere in [R.sup.3] is constructed using the classical method of images and expressed in terms of eigenvalues associated with the surface, leading to an analogue of the Poisson integral as a solution to the Neumann problem for the sphere. In , there are given the Neumann's function and the solution of the Neumann problem for the interior and the exterior of the sphere of [R.sup.n], n [greater than or equal to] 2.

Green functions of the Laplacian for Neumann problems relative to all domains bounded by the coordinate surfaces in the circular cylindrical coordinate system are constructed in .

In  it can be found a construction of the Green function for the three-dimensional Laplace equation, in the interior of an arbitrary rectangular channel subject to homogeneous Neumann conditions on the boundaries.

The explicit forms of the Green's function and of the Neumann's function G and [G.sub.1] in the half-plane [x.sub.2] [greater than or equal to] 0 in [R.sup.2] are given in , while in  there are given the Green's and Neumann's functions for both the half-plane [x.sub.2] [greater than or equal to] 0 and the upper half-space [x.sub.3] [greater than or equal to] 0 in [R.sup.3]. However, in both books it is not proved that the functions u and [u.sub.1] obtained by means of G and [G.sub.1] and Green's formula are solutions of the (DP) and (NP), respectively.

In , the explicit expressions of the Green's and the Neumann's functions are given for the upper half-spaces [R.sup.2.sub.+] and [R.sup.3.sub.+], and it is shown that the functions u and [u.sub.1] obtained with their aid are solutions of the (DP) and (NP), respectively, under the hypothesis that g and [g.sub.1] are analytical. Physical interpretations of the Green's and Neumann's functions can be found in . Also the Green's and Neumann's functions for the interior and the exterior of the unit circle in [R.sup.2] and unit sphere in [R.sup.3] centered at the origin are given in .

In , it is constructed the Green's function for the Neumann problem formulated for the unit sphere in [R.sup.3] centered at the origin. Also in , there are obtained the explicit solutions u and [u.sub.1] of the (DP) and (NP) using Green's formula for the tridimensional upper-half space, respectively (without using the Green's function for (DP) or the Green's function for (NP)). Then it is proved that u and [u.sub.1] are indeed solutions of (DP) and (NP), respectively, under the assumptions: g and [g.sub.1] are continuous on [x.sub.3] = 0, g is bounded and [absolute value of g([x.sub.1], [x.sub.2])] [less than or equal to] M/[([square root of [x.sup.2.sub.1] + [x.sup.2.sub.2]).sup.1+a], where 0 < a < 1, and M is a positive constant.

In , it is derived the Green's function and it is shown a Poisson Formula for the positive half-space of [R.sup.n], n [greater than or equal to] 2.

We obtain the Green type function for the positive half-space [R.sup.n.sub.+], n [greater than or equal to] 2, and use it to find the solution of the (NP) under the assumption that g is continuous and bounded on [R.sup.n-1], and [g.sub.1] is continuous and with compact support in [R.sup.n-1] (subsection 3.3). The cases where [OMEGA] is the positive half space [R.sup.2.sub.+] and [R.sup.3.sub.+], respectively, are presented in details under hypotheses different from the ones considered by other authors (subsection 3.1 and subsection 3.2).

2 Green Functions for Dirichlet and Neumann Problems

Suppose that the function u [member of] [C.sup.2]([OMEGA]) [intersection] [C.sup.1]([bar.[OMEGA]]). Then the following (Riemann-Green) formula holds, i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

where G(x,y) = [psi]([parallel] x - y [parallel]) + [??](x), x [member of] [bar.[OMEGA]], y [member of] [OMEGA], x = y, x [not equal to] y, [??](x) is an arbitrary harmonic function in [OMEGA] and

[psi](r) = [r.sup.2-n]/(2 - n)[[sigma].sub.n], if n > 2 and [psi](r) = 1/2[pi] r, if n = 2, (3')

with r = [parallel]x - y[parallel]. Finally, [[sigma].sub.n] is the surface area of the unit sphere in [R.sup.n]. Suppose that the function G(x, y) in the above formula satisfies the additional property

G(x, y) = 0 for x [member of] [partial derivative][OMEGA], y [member of] [OMEGA]. (4)

Then the solution u of (DP) with the regularity u [member of] [C.sup.2]([OMEGA]) [intersection] [C.sup.1]([bar.[OMEGA]]) (if any) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

A Green's function G(x, y) for [DELTA] on [OMEGA] is a function G as above, i.e., having the properties x [right arrow] G(x,y) belongs to [C.sup.2]([bar.[OMEGA]] \ {y}), [[DELTA].sub.x] G(x, y) = 0 for x [member of] [OMEGA], G(x,y) = 0, for x [member of] [partial derivative][OMEGA], y [member of] [OMEGA].

For the Neumann problem, u is not prescribed on the boundary [partial derivative][OMEGA] of [OMEGA], so the formula (3) suggests to look for a function G = [G.sub.1] with the condition

d[G.sub.1](x,y)/d[[eta].sub.x] = 0, for x [member of] [partial derivative][OMEGA], (6)

instead of G(x, y) = 0, for x [member of] [partial derivative][OMEGA] for the Dirichlet problem. This means to find a function [??](x) = K(x, y) with x [right arrow] K(x, y) in [C.sup.1]([OMEGA]) [intersection] [C.sup.2]([OMEGA]), [for all]y [member of] [OMEGA],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [psi] is given above, i.e., [psi]'(r) = [r.sup.1-n]/[[sigma].sub.n]. Therefore, the solution [u.sub.1] [member of] [C.sup.2]([OMEGA]) [intersection] [C.sup.1]([bar.[OMEGA]]) (if any) of the (NP) is necessarily given by

[u.sub.1](y) = [[integral].sub.[OMEGA]] [G.sub.1](x,y)f(x)dx - [[integral].sub.[partial derivative][OMEGA]] [G.sub.1] (x, y)[g.sub.1](x)d[[sigma].sub.x]. (7)

Such a function [G.sub.1] will be called a Green type function for the (NP) on [OMEGA] (also called Green's function for (NP) in , , , , or Neumann's function in , , , or Green's function of the second kind in ).

3 Green function for the Neumann Problem for [R.sup.n.sub.+]

In this section we will build Green's functions for the (DP) and (NP) for the half-space [R.sup.n.sub.+], n [greater than or equal to] 2, using the ideas developed in Section 2. After that, we will check directly that the corresponding representation formulas for the solutions of (DP) and (NP) are valid under appropriate assumptions on g and [g.sub.1].

First we consider the cases [intersection] = 2 and [intersection] = 3 under hypothesis on g1 different from the ones used by other authors.

3.1 Construction of the Green function for the Neumann Problem for [R.sup.2.sub.+]

 Let [OMEGA] = {x = ([x.sub.1],[x.sub.2]) [member of] [R.sup.2], [x.sub.2]> 0} be the positive half-space. Clearly, [partial derivative][OMEGA] = {x = ([x.sub.1], [x.sub.2]) [member of] [R.sup.2], [x.sub.2] = 0} = {x = ([x.sub.1], 0), [x.sub.1] [member of] R}. For y = ([y.sub.1], [y.sub.2]) define by reflection [y.sup.*] = ([y.sub.1], - [y.sub.2]). Then the function:

G(x,y) = 1/2[pi]-(ln[parallel]x - y[parallel] - ln[parallel]x - [y.sup.*][parallel]), x [member of] [bar.[OMEGA]], y [member of] [OMEGA], x [not equal to] y, (8)

is a Green function for the (DP), and

[G.sub.1](x,y) = 1/2[pi] (ln[parallel]x - y[parallel] + ln[parallel]x - [y.sup.*][parallel]), x [member of] [bar.[OMEGA]], y [member of] [OMEGA], x [not equal to] y, (9)

is a Green type function for (NP).

As [parallel]x - y[parallel] = [parallel]x - [y.sup.*][parallel] = r for y [member of] [OMEGA], x [member of] [partial derivative][OMEGA], it follows that G(x, y) = 0, x [member of] d [OMEGA], y [member of] [OMEGA].

The outward normal [[eta].sub.x] to [partial derivative][OMEGA] = {x = ([x.sub.1], [x.sub.2]) [member of] [R.sup.2], [x.sub.2] = 0} at x [member of] [partial derivative][OMEGA] is [[eta].sub.x] = (0, -1).

The normal derivative of G is:

dG/d[[eta].sub.x] = - [partial derivative]G/[partial derivative][x.sub.2] = [[pi].sup.-1] [y.sub.2][r.sup.-2] with x = ([x.sub.i], 0), y = ([y.sub.i], [y.sub.2]), [y.sub.2] > 0,

where [r.sup.2] = [([x.sub.i] - [y.sub.i]).sup.2] + [y.sup.2.sub.2]. Similarly we can check that d[G.sub.1]/d[[eta].sub.x] = 0.

The formula (3) suggests that a solution to the problem: [DELTA]u = 0 in [OMEGA], u = g on [partial derivative][OMEGA], could be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

with x = ([x.sub.1], 0), g([x.sub.1], 0) = g([x.sub.1]), y = ([y.sub.1], [y.sub.2]), [y.sub.2] > 0.

Clearly [G.sub.1](x, y) = 1/[pi] ln[parallel]x - y[parallel], for x = ([x.sub.1], 0), y [member of] [OMEGA]. Therefore, (3) suggests that a possible solution to the Neumann problem (with f = 0) could be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11)

with x = ([x.sub.1], 0), [g.sub.1]([x.sub.1], 0) = [g.sub.1]([x.sub.1]), y = ([y.sub.1], [y.sub.2]), [y.sub.2] > 0.

The functions g and g must guarantee the convergence of the improper integrals (10) and (11), respectively (i.e., the existence of u and [u.sub.1]), and the fact that these functions u and u are solutions of the above Dirichlet and Neumann problems with f = 0. An important case in which these requirements are fulfilled is given by:

Theorem 1 Let g be continuous and bounded on R and [g.sub.1] be continuous with compact support in R. Then the functions u and [u.sub.1] given by the improper integrals (10) and (11) satisfy:

1. u [member of] [C.sup.2]([OMEGA]) [intersection] C([bar.[OMEGA]])), [DELTA]u = 0 in [OMEGA], u = g on [partial derivative][OMEGA].

2. [u.sub.1] [member of] [C.sup.2]([OMEGA]) [intersection] [C.sup.1]([OMEGA]), [DELTA][u.sub.1] = 0 in [OMEGA], d[u.sub.1]/d[[eta].sub.x] = [g.sub.1], on [partial derivative][OMEGA].

Proof. The key fact is the elementary formula:

[[pi].sup.-1] [[integral].sup.[infinity].sub.-[infinity]] [y.sub.2][dx.sub.1]/[([x.sub.1], [y.sub.1]).sup.2] + [y.sup.2.sub.2] = 1, x ([x.sub.1], 0), y = ([y.sub.1], [y.sub.2]), [y.sub.2] > 0. (12)

which follows by the change of variable: [x.sub.1] - [y.sub.1] = [y.sub.2]z, so [dx.sub.1] = [y.sub.2][d.sub.z].

The fact that u and [u.sub.1] are harmonic in [OMEGA] follows by differentiating under the integral sign, in conjunction with:

[[DELTA].sub.y] [d.sub.G]/d[[eta].sub.x] (x, y) = d/d[[eta].sub.x] ([[DELTA].sub.y]G(x, y)), G(x, y) = G(y, x) for x [member of] [bar.[OMEGA]], y [member of] [OMEGA], x [not equal to] y, and [[DELTA].sub.y] G(x, y) = [[DELTA].sub.y] G(y, x) = 0 for x [member of] [partial derivative][OMEGA], y [member of] [OMEGA].

As g is bounded and [g.sub.1] is continuous and with compact support in R, u and [u.sub.1] given by (10) and (11) are bounded.

Let us to prove that u(x) = g(x) for x = (xi, 0), i.e., that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

In view of (10), we can write

u(y) - g ([y.sub.0]) = [[pi].sup-1] [[integral].sup.[infinity].sub.-[infinity]] [y.sub.2](g([x.sub.1]) - g(y.sup.1.aub.0))[dx.sub.1]/[([x.sub.1] - [y.sub.1]).sup.2] + [y.sup.2.sub.2], (14)

y = ([y.sub.1], [y.sub.2]), [y.sub.2] > 0.

Given [member of] > 0, there exists [delta]([member of]) > 0 such that

[absolute value g([x.sub.1]) - g([y.sup.1.sub.0])] < [member of], for [absolute value of [x.sub.1] - [y.sup.1.sub.0]] < [delta].

Let us write:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

In view of (10) and (12), it follows that [absolute value [I.sub.1]] [less than or equal to] [member of]. Moreover, by the same change of variable as above, we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Similarly,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], as g is bounded on R. This in conjunction with [absolute value [I.sub.1]] [less than or equal to] e implies [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], so (13) is valid.

Under the hypotheses on [g.sub.1], the function [u.sub.1] is well defined and of class [C.sup.1]([bar.[OMEGA]]). Differentiating under the integral sign we obviously obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

On the basis of the previous discussion, this implies d[u.sub.1]/d[[eta].sub.y] = - [partial derivative][u.sub.1] (y)/[partial derivative][y.sub.2] = [g.sub.1](y) on the boundary [partial derivative][OMEGA] of [OMEGA] as [g.sub.1] is bounded, which completes the proof.

3.2 Construction of the Green function for the Neumann Problem for [R.sup.3.sub.+]

We build the Green functions G and [G.sub.1] for the (DP) and (NP), respectively, in the situation where [OMEGA] = [R.sup.3.sub.+], and then we verify directly that the representation formulas of the functions u and [u.sub.1] obtained by means of G and G0 do indeed provide us with solutions of the (DP) and (NP), respectively.

The following classical result will be needed to achieve that (Theorem 1.1.11, ).

Theorem 2  Let f : [R.sup.n] [right arrow] (- [infinity], [infinity] to) be a measurable function and let R > 0.

i) If [absolute value of f (x)] [less than or equal to] c [[parallel]x[parallel].sup.-[lambda]], for all x [member of] [R.sup.n] with [parallel]x[parallel] [less than or equal to] R, where c is a positive constant, and [lambda] < n, then [[integral].sub.[parallel]x[parallel][less than or equal to]R] [absolute value of f(x)]dx < + [infinity].

ii) If [absolute value of f(x)] [less than or equal to] c[[parallel]x[parallel]-[lambda]], for all x [member of] [R.sup.n] with [parallel]x[parallel] [greater than or equal to] R, where c is a positive constant, and [lambda] > n, then [[integral].sub.[parallel]x[parallel][greater than or equal to]R] [absolute value of f (x)] dx < + [infinity].

Next we construct the Green function for the Neumann problem for the positive half-space [R.sup.3.sub.+].

Let [OMEGA] = {x = ([x.sub.1], [x.sub.2], [x.sub.3]) [member of] [R.sup.3], [x.sub.3] > 0} be the positive half-space. Clearly, [partial derivative][OMEGA] = {x = ([x.sub.1], [x.sub.2], [x.sub.3]) [member of] [R.sup.3], [x.sub.3] = 0} = {x = ([x.sub.1], [x.sub.2], 0), ([x.sub.1], [x.sub.2]) [member of] [R.sup.2]}. For y = ([y.sub.1], [y.sub.2], [y.sub.3]) define by reflection [y.sup.*] = ([y.sub.1], [y.sub.2] - [y.sub.3]). Then the function:

G(x,y) = 1/4[pi][parallel]x - y[parallel] - 1/4[pi][parallel]x - [y.sup.*][parallel], x [member of] [bar.[OMEGA]], y [member of] [OMEGA], x [not equal to] y, (16)

is a Green function for the (DP) with f = 0, and

[G.sub.1] (x, y) = 1/4[pi][parallel]x - y[parallel] + 1/4[pi][parallel]x - [y.sup.*][parallel], x [member of] [bar.[OMEGA]], y [member of] [OMEGA], x [not equal to] y, (17)

is a Green type function for (NP) with f = 0.

As [parallel]x - y[parallel] = [parallel]x - [y.sup.*][parallel] = r for y [member of] [OMEGA], x [member of] [partial derivative][OMEGA], it follows that G(x, y) = 0, x [member of] [partial derivative][OMEGA], y [member of] [OMEGA].

The outward normal [[eta].sub.x] to [partial derivative][OMEGA] = {x = ([x.sub.1], [x.sub.2], [x.sub.x3]) [member of] [R.sup.3,] [x.sub.3] = 0} at x [member of] [partial derivative][OMEGA] is [[eta].sub.x] = (0, 0, -1). The normal derivative of G is dG/d[[eta].sub.x] = [partial derivative]G/[partial derivative][x.sub.3] = [y.sup.3]/2[pi][r.sup.3] with x = ([x.sub.1], [x.sub.2], 0), y = ([y.sub.1], [y.sub.2], [y.sub.3]), [y.sub.3] > 0, where [r.sup.2] = [([x.sub.1] - [y.sub.1]).sup.2] +[ ([x.sub.2] - [y.sub.2]).sup.2] + [y.sup.2.sub.3]. Similarly we can check that d[G.sub.1]/d[[eta].sub.x] = 0, for x [member of] [partial derivative][OMEGA], y [member of] [OMEGA].

The formula (3) suggests that a solution to the problem: Am = 0 in [OMEGA], u = g on [partial derivative][OMEGA], could be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (18)

Clearly [G.sub.1](x, y) = 1/2[pi][parallel]x - y[parallel], for x = ([x.sub.1], [x.sub.2] ,0), y [member of] [OMEGA]. Therefore, (3) suggests that a possible solution to the problem [DELTA]u = 0 in [OMEGA], du/d[[eta].sub.x] [g.sub.1] on [partial derivative][OMEGA], could be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (19)

with x = ([x.sub.1], [x.sub.2], 0), [g.sub.1] ([x.sub.1], [x.sub.1], 0) = [g.sub.1] ([x.sub.1], [x.sub.2]), y = ([Y.sub.1], [Y.sub.2], [y.sub.3]), [y.sub.3] > 0.

The functions g and [g.sub.1] must guarantee the convergence of the improper integrals (18) and (19), respectively (i.e., the existence of u and [u.sub.1]), and the fact that these functions u and [u.sub.1] are solutions of the above Dirichlet and Neumann problems with f = 0. An important case in which these requirements are fulfilled is given by:

Theorem 3 Let g be continuous and bounded on [R.sup.2], and [g.sub.1] be continuous and with compact support in [R.sup.2]. Then the functions u and [u.sub.1] given by the improper integrals (18) and (19) satisfy:

1. u [member of] [C.sup.2]([OMEGA]) [intersection] C([bar.[OMEGA]])), [DELTA]u = 0 in [OMEGA], u = g on [partial derivative][OMEGA]

2. [u.sub.1] [member of] [C.sup.2]([OMEGA]) [intersection] [C.sup.1][OMEGA]([bar.[OMEGA]]), [DELTA][u.sub.1] = 0 in [OMEGA], d[u.sub.1]/d[[eta].sub.x] = [g.sub.1] on [partial derivative][OMEGA].

Proof. We will use the formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (20)

where x = ([x.sub.1], [x.sub.2], 0), y = ([y.sub.1], [y.sub.2], [y.sub.3] > 0.

To show (20), we make the change of variables [x.sub.1] - [y.sub.1] = [y.sub.3] z, [x.sub.2] - [y.sub.2] = [y.sub.3] w, so [dx.sub.1] = [y.sub.3] dz, [dx.sub.2] = [y.sub.3] dw. The integral [I.sub.0] becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [integral] dt/[([t.sup.2] + [a.sup.2]).sup.3/2] = t/[a.sup.2][square root of [t.sup.2] + [a.sup.2]] + C, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The fact that u and [u.sub.1] are harmonic on [OMEGA] follows by differentiating under the integral sign and taking into account that [G.sub.1](x, y) = [G.sub.1](y, x), x [member of] [bar.[OMEGA]], y [member of] [OMEGA], x [not equal to] y, [[DELTA].sub.y][G.sub.1](x, y) = [[DELTA].sub.y][G.sub.1](y, x) = 0, y [member of] [OMEGA], x [member of] [partial derivative][OMEGA], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As g is bounded on [R.sup.2], and [g.sub.1] is continuous and with compact support in [R.sup.2], u and [u.sub.1] given by (18) and (19) are bounded due to Theorem 2.

Let us to prove that u(x) = g(x) for x = ([x.sub.1], [x.sub.2], 0), i.e., that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (21)

for all [bar.y] = ([[bar.y].sub.1], [[bar.y].sub.2], 0), y = ([y.sub.1], [y.sub.2], [y.sub.3]), with [y.sub.3] > 0.

In view of (18) and (20), we can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (22)

where y = ([y.sub.1], [y.sub.2], [y.sub.3]), [y.sub.3] > 0, [bar.y] = ([[bar.y].sub.1], [[bar.y].sub.2], 0), x = ([x.sub.1], [x.sub.2], 0), ([x.sub.1], [x.sub.2]) [member of] [R.sup.2].

Given [epsilon] > 0, there exists [delta]([epsilon]) > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (23)

From (20), it follows that [absolute value of [I.sub.1]] [less than or equal to] [epsilon].

We will show that the integral [I.sub.2] converges to zero for y [right arrow] [bar.y].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since g is bounded, let M > 0 be a constant such that [absolute value of g(x)] [less than or equal to] M, for all x [member of] [R.sup.2].

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

as y [right arrow] [bar.y], i.e., as [y.sub.3] [right arrow] 0+, because the above integral is bounded in view of Theorem 2.

We conclude that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which in conjunction with [absolute value of [I.sub.1]] [less than or equal to] [epsilon] implies [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for all [epsilon] > 0, so (21) is valid.

Under the hypotheses on [g.sub.1] the function [u.sub.1] is well defined and of class [C.sup.1] ([bar.[OMEGA]]). Differentiating under the integral sign we obviously obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

On the basis of the previous discussion, this implies:

[du.sub.1]/d[[eta].sub.y] = [partial derivative][u.sub.1](y)/[partial derivative][y.sub.3] = [g.sub.1](y) on the boundary [partial derivative][OMEGA] of [OMEGA] as [g.sub.1] is bounded, which completes the proof.

3.3 Construction of the Green function for the Neumann Problem for [R.sup.n.sub.+]

Next we generalize the result of section 3.2 by constructing the Green function for the Neumann problem for the positive half-space [R.sup.n.sub.+], n > 2.

Let [OMEGA] = [R.sup.n.sub.+] = {x = ([x.sub.1], [x.sub.2], ..., [x.sub.n]) [member of] [R.sup.n], [x.sub.n] > 0}, n > 2, be the positive half-space. Clearly, [partial derivative][OMEGA] = {x = ([x.sub.1], [x.sub.2], ..., [x.sub.n]) [member of] [R.sup.n], [x.sub.n] = 0} = {x = ([x.sub.1], ..., [x.sub.n-1], 0), ([x.sub.1], ..., [x.sub.n-1]) [member of] [R.sup.n-1]}.

For y = ([y.sub.1], [y.sub.2], ..., [y.sub.n]) define by reflection [y.sup.*] = ([y.sub.1], ..., [y.sub.n-1], - [y.sub.n]).

Then the function:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (24)

is a Green function for the (DP) with f = 0, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (25)

is a Green type function for (NP) with f = 0.

As [parallel]x - y[parallel] = [parallel]x - [y.sup.*][parallel] = r for y [member of] [OMEGA], x [member of] [partial derivative][OMEGA], it follows that G(x, y) = 0, x [member of] [partial derivative][OMEGA], y [member of] [OMEGA].

The outward normal [[eta].sub.x] to [partial derivative][OMEGA] at x [member of] [partial derivative][OMEGA] is [[eta].sub.x] = (0, ..., 0, -1). The normal derivative of G is dG/d[[eta].sub.x] = -[partial derivative]G/[partial derivative][x.sub.n] = 2[y.sub.n]/[[sigma].sub.n][r.sub.n], x = ([x.sub.1], ..., [x.sub.n-1], 0), y = ([y.sub.1], ..., [y.sub.n-1], [y.sub.n]), [y.sub.n] > 0, where [r.sup.2] = [([x.sub.1] - [y.sub.1]).sup.2] + ... + [([x.sub.n-1] - [y.sub.n-1]).sup.2] + [y.sup.2.sub.n]. Similarly we can check that [dG.sub.1]/d[[eta].sub.x] = 0, for x [member of] [partial derivative][OMEGA], y [member of] [OMEGA].

The formula (3) suggests that a solution to the problem: [DELTA]u = 0 in [OMEGA], u = g on [partial derivative][OMEGA], could be:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (26)

The function u is the solution of the (DP) with f = 0 (see  for the proof of the Poisson's formula for the half-space of [R.sup.n.sub.+], n > 2).

Clearly [G.sub.1](x, y) = 2/(n - 2)[[sigma].sub.n][[parallel]x - y[parallel].sup.n-2], for x = ([x.sub.1], ..., [x.sub.n-1], 0), y [member of] [R.sup.n.sub.+].

Therefore, (3) suggests that a possible solution to the problem [DELTA]u = 0 in [OMEGA], du/d[[eta].sub.x] = [g.sub.1] on [partial derivative][OMEGA], could be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (27)

The function [g.sub.1] must guarantee the convergence of the improper integral (27) (i.e., the existence of [u.sub.1]), and the fact that this function [u.sub.1] is solution of the Neumann problems with f = 0. An important case in which these requirements are fulfilled is given by:

Theorem 4 Let [g.sub.1] be continuous and with compact support in [R.sup.n-1].

Then the function [u.sub.1] given by the improper integral (27) satisfies:

[u.sub.1] [member of] [C.sup.2]([OMEGA]) [intersection] [C.sup.1]([bar.[OMEGA]]), [DELTA][u.sub.1] = 0 in [OMEGA], [du.sub.1]/d[[eta].sub.x] = [g.sub.1] on [partial derivative][OMEGA].

Proof. We will use the formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (28)

where x = ([x.sub.1], ..., [x.sub.n-1], 0), y = ([y.sub.1], ..., [y.sub.n]), [y.sub.n] > 0.

The fact that u[OMEGA] are harmonic on [OMEGA] follows by differentiating under the integral sign and taking into account that [G.sub.1](x, y) = [G.sub.1](y, x), x [member of] [bar.[OMEGA]], y [member of] [OMEGA], x [not equal to] y, [[DELTA].sub.y][G.sub.1](x, y) = [[DELTA].sub.y][G.sub.1](y, x) = 0, y [member of] [OMEGA], x [member of] [partial derivative][OMEGA].

As [g.sub.1] is continuous and with compact support in [R.sup.n-1], [u.sub.1] defined by (27) is bounded due to Theorem 2.

Let us to prove that du/d[[eta].sub.x] = [g.sub.1](x) for x = ([x.sub.1], ..., [x.sub.n-1], 0), i.e., that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (29)

for all [bar.y] = ([[bar.y].sub.1], ..., [[bar.y].sub.n-1], 0) fixed, and y = ([y.sub.1], ..., [y.sub.n]), [y.sub.n] > 0.

Under the hypotheses on [g.sub.1] the function [u.sub.1] is well defined and of class [C.sup.1]([bar.[OMEGA]]). Differentiating under the integral sign we obviously obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In view of (27) and (28), we can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Given [epsilon] > 0, there exists [delta]([epsilon]) > 0 such that

[absolute value of [g.sub.1](x) - [g.sub.1]([bar.y])] < [epsilon], for [parallel]x - [bar.y][parallel] < [delta].

Then if [parallel]y - [bar.y][parallel] < [delta]/2, y [member of] [R.sup.n.sub.+], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (30)

where B([bar.y], [delta]) = {x [member of] [R.sup.n]; [parallel]x - [bar.y][parallel] < [delta]}.

From (28), it follows that [absolute value of [I.sub.1]] [less than or equal to] [epsilon].

If [parallel]y - [bar.y][parallel] < [delta]/2, [parallel]x - [bar.y][parallel] [greater than or equal to] [delta], we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and so [parallel]x - y[parallel] [greater than or equal to] 1/2 [parallel]x - [bar.y][parallel].

Since [g.sub.1] is bounded, let M > 0 be a constant such that [absolute value of [g.sub.1](x)] [less than or equal to] M, for all x [member of] [R.sup.n-1].

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

as y [right arrow] [bar.y], i.e., as [y.sub.n] [right arrow] 0+, because the above integral is bounded in view of Theorem 2.

We conclude that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which in conjunction with [absolute value of [I.sub.1]] [less than or equal to] [epsilon] implies [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for all [epsilon] > 0, so (29) is valid, which completes the proof.

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