We fill a gap in the theory of elliptic systems on bounded domains, by proving the independence of the index and nullspace 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,
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 welldefined 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 LopatinskiiSchapiro 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 nullspace 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 LopatinskiiSchapiro (LS) condition is probably the combination of proper ellipticity and of the socalled "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 byproduct 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 wellknown 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 lowerorder 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 lowerorder 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 pullback 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 normclose 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 (finitedimensional) 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 nullspace 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
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 orientationpreserving 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 orientationpreserving, 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 boundarypreserving 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 DouglisNirenberg 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 DouglisNirenberg 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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