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Conjugate harmonic function has the form of an equation of the
streamlines (see. figure).
v
(
x, y
) =
ϕ
1
−
ϕ
2
2
π
ln
ch (
πx/
2
a
) + cos (
πy/
2
a
)
ch (
πx/
2
a
)
−
cos (
πy/
2
a
)
,
Conclusion.
The obtained formulae for the solution to a mixed
boundary value problem for the Laplace equation in an infinite layer can
also be applied when the boundary values are the tempered distributions.
In this case the solution is a generalized one, i.e.
u
(
x, y
)
and
u
y
(
x, y
)
when
y
→
+0
and thus
y
→
a
−
0
converge weakly in the space
S
0
(
R
n
)
to the
given boundary values. In case of two variables the specified formulae can
be used to solve plane problems of the underground fluid dynamics (the
theory of filtration).
REFERENCES
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Nauki i Tekhniki. Ser. Sovrem. Probl. Mat.: Fund. Napr.
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E.Yu., Kopaev A.V. On the solution of the Dirichlet problem for some
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