Boundary Value Problems volume 2009, Article number: 637243 (2009)
We fill a gap in the 
theory of elliptic systems on bounded domains, by proving the 
-independence of the index and null-space under "minimal" smoothness assumptions. This result has been known for long if additional regularity is assumed and in various other special cases, possibly for a limited range of values of 
. Here, 
-independence is proved in full generality.
Although important issues are still being investigated today, the bulk of the Fredholm theory of linear elliptic boundary value problems on bounded domains was completed during the 1960s. (For pseudodifferential operators, the literature is more recent and begins with the work of Boutet de Monvel [1]; see also [2] for a more complete exposition.) While this was the result of the work and ideas of many, the most extensive treatment in the 
framework is arguably contained in the 1965 work of Geymonat [3]. This note answers a question explicitly left open in Geymonat's paper which seems to have remained unresolved.
We begin with a brief partial summary of [3] in the case of a single scalar equation. Let 
be a bounded connected open subset of 
, 
, and let 
denote a differential operator on 
of order 
, 
with complex coefficients,
Next, let 
be a system of boundary differential operators on 
with 
of order 
also with complex coefficients,
With 
and 
denoting a chosen integer, introduce the following regularity hypotheses:
(H1; )
is a 
-submanifold of 
(i.e., 
is a 
submanifold of 
and 
lies on one side of 
);
(H2; ) the coefficients 
are in 
if 
and in 
otherwise;
(H3; ) the coefficients 
are of class 
if 
and in 
otherwise.
Then, for 
, the operator 
maps continuously 
into 
and 
maps continuously 
into 
for every 
is a well-defined bounded linear operator. Geymonat's main result [3, Teorema 3.4 and Teorema 3.5] reads as follows.
Theorem 1.1.
Suppose that (H1; 
), (H2; 
), and (H3; 
) hold for some 
Then,
(i)if 
and 
the operator 
is Fredholm if and only if 
is uniformly elliptic in 
and 
satisfies the Lopatinskii-Schapiro condition (see below);
(ii)if also 
and 
is Fredholm for some 
and some 
(and hence for every such 
and 
by (i), both the index and null-space of 
are independent of 
and 
.
The assumptions made in Theorem 1.1 are nearly optimal. In fact, most subsequent expositions retain more smoothness of the boundary and leading coefficients to make the parametrix calculation a little less technical.
The best known version of the Lopatinskii-Schapiro (LS) condition is probably the combination of proper ellipticity and of the so-called "complementing condition." Since we will not use it explicitly, we simply refer to the standard literature (e.g., [3–5]) for details.
We will fill the obvious "gap" between (i) and (ii) of Theorem 1.1 by proving what follows.
Theorem 1.2.
Theorem 1.1(ii) remains true if 
Note that 
corresponds to the most general hypotheses about the boundary and the coefficients, which is often important in practice.
From now on, we set 
for simplicity of notation. The reason why 
is required in part (ii) of Theorem 1.1 is that the proof uses part (i) with 
replaced by 
By a different argument, a weaker form of Theorem 1.2 was proved in [3, Proposizione 4.2] (
-independence for 
in some bounded open interval around the value 
under additional technical conditions).
If 
is invertible for some 
and every 
then Theorem 1.2 is a straightforward by-product of the Sobolev embedding theorems and, in fact, 
in this case. However, this invertibility can only be obtained under more restrictive ellipticity hypotheses (such as strong ellipticity) and/or less general boundary conditions (Agmon [6], Browder [7], Denk et al. [8, Theorem 
, page 102]).
The idea of the proof of Theorem 1.2 is to derive the case 
from the case 
by regularization of the coefficients and stability of the Fredholm index. The major obstacle in doing so is the mere 
regularity of 
since Theorem 1.1 with 
can only be used if 
is 
or better. This will be overcome in a somewhat nonstandard way in these matters, by artificially increasing the smoothness of the boundary with the help of the following lemma.
Lemma 1.3.
Suppose that 
is a bounded open subset of 
and that 
is a 
-submanifold of 
of class 
with 
Then, there is a bounded open subset 
of 
such that 
is a 
-submanifold of 
of class 
(even 
) and that 
and 
are 
diffeomorphic (as 
-manifolds).
The next section is devoted to the (simple) proof of Theorem 1.2 based on Lemma 1.3 and to a useful equivalent formulation (Corollary 2.1). Surprisingly, we have been unable to find any direct or indirect reference to Lemma 1.3 in the classical differential topology or PDE literature. It does not follow from the related and well-known fact that every 
-manifold 
of class 
with 
is 
diffeomorphic to a 
-manifold 
of class 
since this does not ensure that both can always be embedded in the same euclidian space. It is also clearly different from the results just stating that 
can be approximated by open subsets with a smooth boundary (as in [9]), which in fact need not even be homeomorphic to 
Accordingly, a proof of Lemma 1.3 is given in Section 3.
Based on the method of proof and the validity of Theorem 1.1 for systems after suitable modifications of the definition of 
in (1.3) and of the hypotheses (H1; 
), (H2; 
), and (H3; 
), there is no difficulty in checking that Theorem 1.2 remains valid for most systems as well, but a brief discussion is given in Section 4 to make this task easier.
Remark 1.4.
When the boundary 
is not connected, the system 
of boundary conditions may be replaced by a collection of such systems, one for each connected component of 
Theorems 1.1 and 1.2 remain of course true in that setting, with the obvious modification of the target space in (1.3).
As noted in [3, page 241], the 
-independence of 
(recall 
) follows from that of 
so that it will suffice to focus on the latter.
The problem can be reduced to the case when the lower-order coefficients in 
and 
vanish since the operator they account for is compact from the source space to the target space in (1.3), irrespective of 
Thus, the lower-order terms have no impact on the existence of 
or on its value. It is actually more convenient to deal with the intermediate case when all the coefficients 
are in 
and all the coefficients 
are in 
which is henceforth assumed.
First, 
since 
and 
so that by (H1; 
) and Lemma 1.3, there are a bounded open subset 
of 
such that 
is a 
-submanifold of 
of class 
and a 
diffeomorphism 
mapping 
onto 
The pull-back 
is a linear isomorphism of 
onto 
for every 
and of 
onto 
for every 
Meanwhile, 
where 
is a differential operator of order 
with coefficients 
of class 
on 
and 
where 
is a differential operator of order 
with coefficients 
of class 
on 
From the above remarks, the operator (where 
)
has the form 
where 
and 
are isomorphisms. As a result, 
is Fredholm with the same index as 
Since the coefficients of 
and 
and of 
and 
have the same smoothness, respectively, we may, upon replacing 
by 
and 
by 
continue the proof under the assumption that 
is a 
submanifold of 
(but the 
are still 
and the 
still 
).
The coefficients 
can be approximated in 
by coefficients 
and the coefficients 
can be approximated in 
by 
functions 
on 
(since 
is 
see, e.g., [10, Theorem 
, page 49]), which yields operators 
and 
, 
of order 
and 
respectively, in the obvious way.
Let 
be fixed. The corresponding operators 
and 
are arbitrarily norm-close to 
and 
if the approximation of the coefficients is close enough. If so, by the openness of the set of Fredholm operators and the local constancy of the index, it follows that 
and 
are Fredholm with 
and 
But since 
is now 
and the coefficients 
and 
are 
the hypotheses (H1; 
), (H2; 
), and (H3; 
) are satisfied by 
, 
and 
and any 
Thus, 
by part (ii) of Theorem 1.1, so that 
This completes the proof of Theorem 1.2.
Corollary 2.1.
Suppose that (H1; 
), (H2; 
), and (H3; 
) hold, that 
is uniformly elliptic in 
, and that 
satisfies the LS condition. Let 
If 
and 
, then 
Proof.
Since the result is trivial if 
we assume 
Obviously, 
and 
is Fredholm by Theorem 1.1(i). Let 
denote a (finite-dimensional) complement of 
in 
Since 
is dense in 
and 
is closed, we may assume that 
If not, approximate a basis of 
by elements of 
If the approximation is close enough, the approximate basis is linearly independent and its span 
(of dimension 
) intersects 
only at 
(by the closedness of 
). Thus, 
may be replaced by 
as a complement of 
.
Since 
and 
have the same index and null-space by Theorem 1.2, their ranges have the same codimension. Now, 
because 
is a complement of 
and 
This shows that 
is also a complement of 
Therefore, since 
there is 
such that 
This yields 
whence 
and so 
This means that 
for some 
Thus, 
that is, 
Since 
by Theorem 1.2, it follows that 
It is not hard to check that Corollary 2.1 is actually equivalent to Theorem 1.2. This was noted by Geymonat, along with the fact that Corollary 2.1 was only known to be true in special cases ([3, page 242]).
Under the assumptions of Lemma 1.3, 
has a finite number of connected components, each of which satisfies the same assumptions as 
itself. Thus, with no loss of generality, we will assume that 
is connected.
If 
and 
are 
-manifolds of class 
with 
and 
and 
are 
diffeomorphic, they are also 
diffeomorphic ([10, Theorem 
, page 57]). Thus, since 
is of class 
with 
it suffices to find a bounded open subset 
of 
such that 
is 
and 
diffeomorphic to 
In a first step, we find a 
function 
such that 
and 
on 
while 
in 
, 
in 
and 
This can be done in various ways and even when 
However, since 
the most convenient argument is to rely on the fact that the signed distance function
is 
in 
, where 
, and
is an open neighborhood of 
in 
This is shown in Gilbarg and Trudinger [11, page 355] and also in Krantz and Parks [12]. Both proofs reveal that 
when 
that is, when 
(Without further assumptions, the 
regularity of 
breaks down when 
)
Let 
be nondecreasing and such that 
if 
and 
if 
where 
is given. Then, 
is 
in 
vanishes only on 
, and 
on 
Furthermore, since 
on a neighborhood of 
in 
and 
on a neighborhood of 
in 
, 
remains 
after being extended to 
by setting 
if 
and 
if 
This 
satisfies all the required conditions except 
Since 
for 
large enough, this can be achieved by replacing 
by 
Since 
off 
it follows from a classical theorem of Whitney [13, Theorem III] (with 
in that theorem) that there is a 
function 
on 
of class 
in 
such that, if 
then 
if 
and 
if 
Evidently, 
does not vanish on 
and 
has the same sign as 
off 
, that is, 
in 
and 
in 
Furthermore, 
for every 
so that 
for 
for some 
Upon shrinking 
we may assume that 
Also, 
For convenience, we summarize the relevant properties of 
below:
(i)
is 
on 
and 
off 
,
(ii)
for 
,
(iii)
,
(iv)
,
(v)
Choose 
It follows from (v) that 
is compact and, from (iii) and (iv), that 
if 
is small enough (argue by contradiction). Since 
by (iii) and (iv) and since 
this implies 
Thus, by (i) and (ii), 
is a 
submanifold of 
and the boundary of the open set 
In fact, 
is a 
-manifold of class 
since, once again by (ii), 
lies on one side of its boundary.
We now proceed to show that 
is 
diffeomorphic to 
This will be done by a variant of the procedure used to prove that nearby noncritical level sets on compact manifolds are diffeomorphic. However, since we are dealing with sublevel sets and since critical points will abound, the details are significantly different.
Let 
be such that 
and 
on 
Since 
on 
by (ii), the function 
extended by 
outside 
is a bounded 
vector field on 
Since 
the function 
defined by
is well defined and of class 
and 
is an orientation-preserving 
diffeomorphism of 
for every 
We claim that 
produces the desired diffeomorphism from 
to 
It follows at once from (3.3) that 
so that 
is decreasing along the flow lines and hence that 
maps 
into itself for every 
Also, if 
then 
for every 
so that 
by (iii). If now 
then 
and 
is strictly decreasing for 
small enough. It follows that 
that is, 
for 
Altogether, this yields 
Suppose now that 
Then, 
and hence 
For 
small enough, 
and so 
for 
small enough. In fact, it is obvious that 
until 
is large enough that 
But since 
and 
is decreasing along the flow lines, 
implies 
Since 
this means that 
for some 
Call 
the first (and, in fact, only, but this is unimportant) time when 
From the above, 
for 
and hence for 
since 
Then, 
for 
so that 
for 
In particular, since 
and hence 
it follows that 
In other words, 
Thus, 
that is, 
If 
(so that 
and hence 
), this yields 
On the other hand, if 
then 
Since 
, 
is strictly decreasing for 
near 
and so 
whence 
The above shows that 
maps 
into 
, 
into 
, and 
into 
That it actually maps 
onto
follows from a Brouwer's degree argument: 
is connected and no point of 
is in 
since, as just noted, 
Thus, for 
is defined and independent of 
Now, choose 
so that 
Then, 
for every 
and so 
Since 
is one to one and orientation-preserving, it follows that 
and so 
for every 
Thus, there is 
such that 
which proves the claimed surjectivity.
At this stage, we have shown that 
is a 
diffeomorphism of 
mapping 
into 
, 
into 
, and 
into and onto 
It is straightforward to check that such a diffeomorphism also maps 
onto 
(approximate 
by a sequence from 
) and hence it is a boundary-preserving diffeomorphism of 
onto 
This completes the proof of Lemma 1.3.
Remark 3.1.
The 
diffeomorphism 
above is induced by a diffeomorphism of 
but this does not mean that the same thing is true of the 
diffeomorphism of Lemma 1.3.
Suppose now that 
, 
, 
, is a system of 
differential operators on 
which is properly elliptic in the sense of Douglis and Nirenberg [14]. We henceforth assume some familiarity with the nomenclature and basic assumptions of [4, 14]. Recall that Douglis-Nirenberg ellipticity is equivalent to a more readily usable condition due to Volevič [15]. See [5] for a statement and simple proof.
Let 
and 
be two sets of Douglis-Nirenberg numbers, so that 
that have been normalized so that 
and 
It is well known that since 
proper ellipticity implies 
with 
We assume that a system 
, 
, 
of boundary differential operators is given, with 
for some 
Let
and call 
and 
the (complex valued) coefficients of 
and 
respectively. Given an integer 
introduce the following hypotheses (generalizing those for a single equation in the Introduction).
(H1; )
is a 
-submanifold of 
(H2; ) The coefficients 
are in 
if 
and in 
otherwise.
(H3; ) The coefficients 
are in 
if 
and in 
otherwise.
For 
and 
define
Then (as proved in [3]), Theorem 1.1 holds (once again, the LS condition amounts to proper ellipticity plus complementing condition and proper ellipticity is equivalent to ellipticity if 
and 
) and it is straightforward to check that the proof of Theorem 1.2 carries over to this case if 
If so, Corollary 2.1 is also valid, with a similar proof and an obvious modification of the function spaces.
Remark 4.1.
If 
there is no boundary condition (in particular, 
, and (H3; 
) is vacuous) and the system 
can be solved explicitly for 
in terms of 
and its derivatives. This is explained in [14, page 506]. If so, the smoothness of 
(i.e., (H1; 
) is irrelevant, and Theorem 1.2 is trivially true regardless of 
(
is an isomorphism). A special case when 
arises if 
(in particular, if 
), for then 
from the conditions 
and 
From the above, Theorem 1.2 may only fail if 
, 
, and 
. (The author was recently informed by H. Koch [16] that he could prove Lemma 1.3 when 
so that Theorem 1.2 remains true in this case as well.)
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Rabier, P. A Complement to the Fredholm Theory of Elliptic Systems on Bounded Domains. Bound Value Probl 2009, 637243 (2009). https://doi.org/10.1155/2009/637243
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DOI: https://doi.org/10.1155/2009/637243