MATH 220: SOLVING PDES
We now return to solving PDE using duality arguments and energy estimates.
Before getting into details, we note that the ideal kind of well-posedness result
we would like is the following. We are given a PDE (including various additional
conditions), and we would like to have spaces of functions
X
and
Y
such that there
is a unique solution of the PDE in
X
which depends continuously on the data in
Y
. To be speciﬁc, let’s consider the linear PDE
Pu
=
f
, with any other condition
we may want to have made homogeneous (e.g. homogeneous Dirichlet BC) – which
we have seen can be done – and we want to ﬁnd function spaces
X
and
Y
, which
should have a notion of convergence, e.g. be normed spaces, such that
(i) (Existence) For any
f
∈
Y
there exists
u
∈
X
such that
Pu
=
f
.
(ii) (Uniqueness) For any
f
∈
Y
there is at most one
u
∈
X
such that
Pu
=
f
.
(iii) (Stability) If
f
j
→
f
in
Y
then the unique solutions,
u
j
, resp.
u
, in
X
, of
Pu
j
=
f
j
,
Pu
=
f
, satisfy
u
j
→
u
in
X
.
Of course,
Pu
needs to have some meaning, so a minimal requirement is that
X
and
Y
are both subspaces of
D
0
(Ω) for some Ω – e.g. Ω =
R
n
, or Ω a bounded domain
in
R
n
, or Ω =
T
n
= (
S
1
)
n
,
C
∞
functions on which, recall, can be identiﬁed with
2
π
-periodic
C
∞
functions in every coordinate on
R
n
. Also, to make this problem
meaningful,
Y
should be relatively large – typically it should include
C
∞
c
(Ω) at
least. (If
P
is linear and
Y
=
{
0
}
, it is trivial to ﬁnd an
X
:
X
=
{
0
}
will do.)
Now, to get a feeling for what this means, note that if we can solve a PDE in
a space
X
(let’s keep
Y
ﬁxed for now), then we can also solve it in a bigger space
˜
X
⊃
X
, since the solution
u
in
X
also lies in
˜
X
. On the other hand, if there is at
most one solution of the PDE in a space
X
, then there is at most one solution in a
smaller space
˜
X
⊂
X
since if
u
1
and
u
2
are solutions in
˜
X
then they are solutions
in
X
, thus by uniqueness in
X
they are equal. Thus, there is some tension between
existence and uniqueness: for the former, one could try to increase the space to
make the problem easier, but then may lose uniqueness, for the latter one may try
to make the space smaller, but then may lose existence. To be extreme, it may
be easy to show uniqueness if
X
=
C
∞
c
(Ω) (e.g. using the maximum principle or
energy estimates), but existence may not hold, while it may be relatively easy to
show existence when
X
=
D
0
(Ω), but uniqueness may not hold then.
As a concrete example, if
Y
=
C
∞
c
(
R
),
P
=
d
dx
, then with
X
=
C
∞
c
(
R
) one has at
most one solution of
Pu
=
f
,
f
∈
Y
(since by the fundamental theorem of calculus
one can write
u
as the indeﬁnite integral of
f
from
-∞
), but may not have any
solutions (this indeﬁnite integral may not vanish for large
x
), while if
X
=
D
0
(
R
),
or indeed if
X
=
C
∞
(
R
), there is always a solution (the aforementioned indeﬁnite
integral), but it is not unique (one can always add a constant). If however one lets