Solution to the mixed boundary-value problem for Laplace equation in multidimentional infinite layer

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It is easy to notice that kn(|t| , y) and ln ( |t| , y) are infinitely differentiable
and rapidly decreasing functions t ∈ Rn, i.e. they belong to the space
S(Rn) [7]. These functions are also spherically symmetric, so while
calculating the inverse Fourier transform it is possible to pass to the
spherical coordinates in the space R n. Let us denote |x| = r, |t| = ρ, σn−1
as an area of the unit sphere in the space Rn. For any spherically symmetric
function hn (|t|) = hn (ρ) ∈ S ( Rn) we have
Hn ( |x|) = Hn (r) = F −1
t [hn] ( x) =
1
(2π) n Z
Rn
hn (|t|) e−i hx,tidt =
=
σn− 1
(2π )n
∞ Z0
hn (ρ) ρ
n
2 dρ
π Z0
e−irρcos θ sinn−2θdθ =
=
1
(2π )
n
2 r
n
2 −1
∞ Z0
hn ( ρ) ρ
n
2 J n
2 −1 (rρ) dρ,
where J n
2 −1 (rρ) is the Bessel function of the first kind of order ν = n/2−1
[7, 8]. The above formula is valid and it is easy to be checked.
By differentiating the above equation and by considering the formula
of the theory of Bessel functions [4]
νJ ν ( rρ)
rρ − J0
ν ( rρ) = Jν +1 (rρ) ,
we obtain the recurrence formula
Hn+2 (r) = −
1
2πr
∂
∂r
Hn (r ) .
For this formula it is necessary to know the kernels Kn ( x, y) and
Ln (x, y) only for n = 1 and n = 2.
Since the Fourier transform moves the space S (Rn) into itself,
the kernels Kn (x, y) = K∗
n (|x| , y ) and Ln (x, y ) = L∗
n (|x | , y ) when
∀y ∈ (0, a), are spherically symmetric functions of x in the space S (Rn).
Therefore, the convolution (1) exists for any tempered distribution ϕ ( x) ∈
∈ S0 (Rn), ψ (x) ∈ S 0 (R n) , and can be written as
ϕ (x) ∗ Kn (x, y) + ψ (x ) ∗ Ln (x, y) =
= ( ϕ (t) , Kn (x − t, y)) + (ψ (t) , Ln ( x − t, y)) .
When ϕ (x ) and ψ ( x) are normal functions of polynomial growth,
convolution (1) can be written as a sum of integrals
Z
Rn
ϕ ( t) Kn ( x − t, y) dt + Z
Rn
ψ (t) Ln (x − t, y) dt .
6 ISSN 1812-3368. Herald of the BMSTU. Series “Natural Sciences”. 2015. No. 1