86
VIII – Cauchy Theory
11 – The Dirichlet problem for the half-plane
The computations of the previous n
◦ allow the Dirichlet problem
50 solution
method presented in Chapter VII, § 5 for functions on the circle T to be
transposed to the functions defined on R. We could avoid using the Fourier
transform and introduce the Poisson transform (11.3) from the start, but
as I explained in the preface to vol. I, my aim is not to necessarily provide
readers with the most direct paths to interesting results. In fact, introducing
the Fourier transform in this very particular situation is neither more nor less
artificial that using Fourier series in the case of the unit disc; the presence of
a group is exploited: the group of rotations about 0 in the case of the disc,
the group of horizontal translations in the case of the half-plane. The method
could be generalized to heat or wave propagation equations
du/dt = Δu , d
2 u/dt
2 = Δu
(
∗ )
where u(t, x) is a function on R + × R
n with (as well as its partial derivative
du/dt in the second case) given values for t = 0. The reader will already
be able to practice on the case n = 1, the basic idea being the one applied
by Fourier to go from the heat equation on the unit circle to his series: find
the “ simple ” solutions of the form f (t)g(x), then try to express the general
solution as a “ continuous sum ” of such solutions; the calculation is easy for
the first equation, but less so for the second one.
Allowing for notation, in the previous n
◦ , the Fourier transform of the
function
ϕ(t) = t
s−1
+ e(zt) , Im(z) > 0 , Re(s) > 0 ,
has been shown to be
ˆ
ϕ(u) = Γ (s) [2πi(u − z)]
−s .
On the other hand, we know (Chap. VII, § 6, n
◦ 30) that if f, g ∈ L
1 (R), then
f (t)ˆ g(t)dt =
ˆ
f (u)g(u)du;
(
∗ )
this formula is obtained by calculating the double integral
f (t)g(u)e(−tu)dtdu
50 Recall that, generally speaking, it consists in constructing a harmonic function
on an open set with given values on the boundary. The cases of the unit disc
or of the half-plane cannot give any idea of the difficulty involved in the general
problem in C, even less so in R
n – not to mention generalizations to elliptic
PDEs.
VIII – Cauchy Theory
11 – The Dirichlet problem for the half-plane
The computations of the previous n
◦ allow the Dirichlet problem
50 solution
method presented in Chapter VII, § 5 for functions on the circle T to be
transposed to the functions defined on R. We could avoid using the Fourier
transform and introduce the Poisson transform (11.3) from the start, but
as I explained in the preface to vol. I, my aim is not to necessarily provide
readers with the most direct paths to interesting results. In fact, introducing
the Fourier transform in this very particular situation is neither more nor less
artificial that using Fourier series in the case of the unit disc; the presence of
a group is exploited: the group of rotations about 0 in the case of the disc,
the group of horizontal translations in the case of the half-plane. The method
could be generalized to heat or wave propagation equations
du/dt = Δu , d
2 u/dt
2 = Δu
(
∗ )
where u(t, x) is a function on R + × R
n with (as well as its partial derivative
du/dt in the second case) given values for t = 0. The reader will already
be able to practice on the case n = 1, the basic idea being the one applied
by Fourier to go from the heat equation on the unit circle to his series: find
the “ simple ” solutions of the form f (t)g(x), then try to express the general
solution as a “ continuous sum ” of such solutions; the calculation is easy for
the first equation, but less so for the second one.
Allowing for notation, in the previous n
◦ , the Fourier transform of the
function
ϕ(t) = t
s−1
+ e(zt) , Im(z) > 0 , Re(s) > 0 ,
has been shown to be
ˆ
ϕ(u) = Γ (s) [2πi(u − z)]
−s .
On the other hand, we know (Chap. VII, § 6, n
◦ 30) that if f, g ∈ L
1 (R), then
f (t)ˆ g(t)dt =
ˆ
f (u)g(u)du;
(
∗ )
this formula is obtained by calculating the double integral
f (t)g(u)e(−tu)dtdu
50 Recall that, generally speaking, it consists in constructing a harmonic function
on an open set with given values on the boundary. The cases of the unit disc
or of the half-plane cannot give any idea of the difficulty involved in the general
problem in C, even less so in R
n – not to mention generalizations to elliptic
PDEs.