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A Complement to the Fredholm Theory of Elliptic Systems on Bounded Domains
Boundary Value Problems volume 2009, Article number: 637243 (2009)
Abstract
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.
1. Introduction
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_Equ1_HTML.gif)
Next, let be a system of boundary differential operators on
with
of order
also with complex coefficients,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_Equ3_HTML.gif)
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).
2. Proof of Theorem 1.2
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 )
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_Equ4_HTML.gif)
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]).
3. Proof of Lemma 1.3
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_Equ5_HTML.gif)
is in
, where
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_Equ6_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_Equ7_HTML.gif)
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.
4. Systems
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_Equ8_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_Equ9_HTML.gif)
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