A Complement to the Fredholm Theory of Elliptic Systems on Bounded Domains

Boundary Value Problems20092009:637243

DOI: 10.1155/2009/637243

Received: 24 March 2009

Accepted: 11 June 2009

Published: 14 July 2009

Abstract

We fill a gap in the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq1_HTML.gif theory of elliptic systems on bounded domains, by proving the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq2_HTML.gif -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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq3_HTML.gif . Here, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq4_HTML.gif -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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq5_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq6_HTML.gif be a bounded connected open subset of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq7_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq8_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq9_HTML.gif denote a differential operator on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq10_HTML.gif of order http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq11_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq12_HTML.gif with complex coefficients,
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_Equ1_HTML.gif
(11)
Next, let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq13_HTML.gif be a system of boundary differential operators on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq14_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq15_HTML.gif of order http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq16_HTML.gif also with complex coefficients,
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_Equ2_HTML.gif
(12)

With http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq17_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq18_HTML.gif denoting a chosen integer, introduce the following regularity hypotheses:

(H1; ) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq20_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq21_HTML.gif -submanifold of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq22_HTML.gif (i.e., http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq23_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq24_HTML.gif submanifold of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq25_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq26_HTML.gif lies on one side of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq27_HTML.gif );

(H2; ) the coefficients http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq29_HTML.gif are in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq30_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq31_HTML.gif and in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq32_HTML.gif otherwise;

(H3; ) the coefficients http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq34_HTML.gif are of class http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq35_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq36_HTML.gif and in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq37_HTML.gif otherwise.

Then, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq38_HTML.gif , the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq39_HTML.gif maps continuously http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq40_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq41_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq42_HTML.gif maps continuously http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq43_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq44_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq45_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_Equ3_HTML.gif
(13)

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; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq46_HTML.gif ), (H2; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq47_HTML.gif ), and (H3; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq48_HTML.gif ) hold for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq49_HTML.gif Then,

(i)if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq50_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq51_HTML.gif the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq52_HTML.gif is Fredholm if and only if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq53_HTML.gif is uniformly elliptic in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq54_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq55_HTML.gif satisfies the Lopatinskii-Schapiro condition (see below);

(ii)if also http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq56_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq57_HTML.gif is Fredholm for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq58_HTML.gif and some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq59_HTML.gif (and hence for every such http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq60_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq61_HTML.gif by (i), both the index and null-space of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq62_HTML.gif are independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq63_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq64_HTML.gif .

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., [35]) 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq65_HTML.gif

Note that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq66_HTML.gif corresponds to the most general hypotheses about the boundary and the coefficients, which is often important in practice.

From now on, we set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq67_HTML.gif for simplicity of notation. The reason why http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq68_HTML.gif is required in part (ii) of Theorem 1.1 is that the proof uses part (i) with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq69_HTML.gif replaced by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq70_HTML.gif By a different argument, a weaker form of Theorem 1.2 was proved in [3, Proposizione 4.2] ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq71_HTML.gif -independence for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq72_HTML.gif in some bounded open interval around the value http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq73_HTML.gif under additional technical conditions).

If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq74_HTML.gif is invertible for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq75_HTML.gif and every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq76_HTML.gif then Theorem 1.2 is a straightforward by-product of the Sobolev embedding theorems and, in fact, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq77_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq78_HTML.gif , page 102]).

The idea of the proof of Theorem 1.2 is to derive the case http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq79_HTML.gif from the case http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq80_HTML.gif by regularization of the coefficients and stability of the Fredholm index. The major obstacle in doing so is the mere http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq81_HTML.gif regularity of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq82_HTML.gif since Theorem 1.1 with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq83_HTML.gif can only be used if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq84_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq85_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq86_HTML.gif is a bounded open subset of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq87_HTML.gif and that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq88_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq89_HTML.gif -submanifold of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq90_HTML.gif of class http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq91_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq92_HTML.gif Then, there is a bounded open subset http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq93_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq94_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq95_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq96_HTML.gif -submanifold of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq97_HTML.gif of class http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq98_HTML.gif (even http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq99_HTML.gif ) and that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq100_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq101_HTML.gif are http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq102_HTML.gif diffeomorphic (as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq103_HTML.gif -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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq104_HTML.gif -manifold http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq105_HTML.gif of class http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq106_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq107_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq108_HTML.gif diffeomorphic to a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq109_HTML.gif -manifold http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq110_HTML.gif of class http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq111_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq112_HTML.gif can be approximated by open subsets with a smooth boundary (as in [9]), which in fact need not even be homeomorphic to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq113_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq114_HTML.gif in (1.3) and of the hypotheses (H1; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq115_HTML.gif ), (H2; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq116_HTML.gif ), and (H3; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq117_HTML.gif ), 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq118_HTML.gif is not connected, the system http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq119_HTML.gif of boundary conditions may be replaced by a collection of such systems, one for each connected component of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq120_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq121_HTML.gif -independence of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq122_HTML.gif (recall http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq123_HTML.gif ) follows from that of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq124_HTML.gif so that it will suffice to focus on the latter.

The problem can be reduced to the case when the lower-order coefficients in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq125_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq126_HTML.gif vanish since the operator they account for is compact from the source space to the target space in (1.3), irrespective of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq127_HTML.gif Thus, the lower-order terms have no impact on the existence of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq128_HTML.gif or on its value. It is actually more convenient to deal with the intermediate case when all the coefficients http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq129_HTML.gif are in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq130_HTML.gif and all the coefficients http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq131_HTML.gif are in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq132_HTML.gif which is henceforth assumed.

First, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq133_HTML.gif since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq134_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq135_HTML.gif so that by (H1; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq136_HTML.gif ) and Lemma 1.3, there are a bounded open subset http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq137_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq138_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq139_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq140_HTML.gif -submanifold of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq141_HTML.gif of class http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq142_HTML.gif and a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq143_HTML.gif diffeomorphism http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq144_HTML.gif mapping http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq145_HTML.gif onto http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq146_HTML.gif

The pull-back http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq147_HTML.gif is a linear isomorphism of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq148_HTML.gif onto http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq149_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq150_HTML.gif and of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq151_HTML.gif onto http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq152_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq153_HTML.gif Meanwhile, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq154_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq155_HTML.gif is a differential operator of order http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq156_HTML.gif with coefficients http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq157_HTML.gif of class http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq158_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq159_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq160_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq161_HTML.gif is a differential operator of order http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq162_HTML.gif with coefficients http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq163_HTML.gif of class http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq164_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq165_HTML.gif

From the above remarks, the operator (where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq166_HTML.gif )
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_Equ4_HTML.gif
(21)

has the form http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq167_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq168_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq169_HTML.gif are isomorphisms. As a result, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq170_HTML.gif is Fredholm with the same index as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq171_HTML.gif Since the coefficients of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq172_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq173_HTML.gif and of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq174_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq175_HTML.gif have the same smoothness, respectively, we may, upon replacing http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq176_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq177_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq178_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq179_HTML.gif continue the proof under the assumption that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq180_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq181_HTML.gif submanifold of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq182_HTML.gif (but the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq183_HTML.gif are still http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq184_HTML.gif and the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq185_HTML.gif still http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq186_HTML.gif ).

The coefficients http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq187_HTML.gif can be approximated in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq188_HTML.gif by coefficients http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq189_HTML.gif and the coefficients http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq190_HTML.gif can be approximated in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq191_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq192_HTML.gif functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq193_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq194_HTML.gif (since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq195_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq196_HTML.gif see, e.g., [10, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq197_HTML.gif , page 49]), which yields operators http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq198_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq199_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq200_HTML.gif of order http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq201_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq202_HTML.gif respectively, in the obvious way.

Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq203_HTML.gif be fixed. The corresponding operators http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq204_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq205_HTML.gif are arbitrarily norm-close to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq206_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq207_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq208_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq209_HTML.gif are Fredholm with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq210_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq211_HTML.gif But since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq212_HTML.gif is now http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq213_HTML.gif and the coefficients http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq214_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq215_HTML.gif are http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq216_HTML.gif the hypotheses (H1; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq217_HTML.gif ), (H2; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq218_HTML.gif ), and (H3; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq219_HTML.gif ) are satisfied by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq220_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq221_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq222_HTML.gif and any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq223_HTML.gif Thus, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq224_HTML.gif by part (ii) of Theorem 1.1, so that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq225_HTML.gif This completes the proof of Theorem 1.2.

Corollary 2.1.

Suppose that (H1; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq226_HTML.gif ), (H2; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq227_HTML.gif ), and (H3; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq228_HTML.gif ) hold, that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq229_HTML.gif is uniformly elliptic in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq230_HTML.gif , and that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq231_HTML.gif satisfies the LS condition. Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq232_HTML.gif If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq233_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq234_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq235_HTML.gif

Proof.

Since the result is trivial if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq236_HTML.gif we assume http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq237_HTML.gif Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq238_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq239_HTML.gif is Fredholm by Theorem 1.1(i). Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq240_HTML.gif denote a (finite-dimensional) complement of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq241_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq242_HTML.gif Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq243_HTML.gif is dense in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq244_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq245_HTML.gif is closed, we may assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq246_HTML.gif If not, approximate a basis of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq247_HTML.gif by elements of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq248_HTML.gif If the approximation is close enough, the approximate basis is linearly independent and its span http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq249_HTML.gif (of dimension http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq250_HTML.gif ) intersects http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq251_HTML.gif only at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq252_HTML.gif (by the closedness of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq253_HTML.gif ). Thus, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq254_HTML.gif may be replaced by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq255_HTML.gif as a complement of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq256_HTML.gif .

Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq257_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq258_HTML.gif have the same index and null-space by Theorem 1.2, their ranges have the same codimension. Now, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq259_HTML.gif because http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq260_HTML.gif is a complement of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq261_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq262_HTML.gif This shows that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq263_HTML.gif is also a complement of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq264_HTML.gif

Therefore, since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq265_HTML.gif there is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq266_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq267_HTML.gif This yields http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq268_HTML.gif whence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq269_HTML.gif and so http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq270_HTML.gif This means that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq271_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq272_HTML.gif Thus, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq273_HTML.gif that is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq274_HTML.gif Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq275_HTML.gif by Theorem 1.2, it follows that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq276_HTML.gif

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, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq277_HTML.gif has a finite number of connected components, each of which satisfies the same assumptions as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq278_HTML.gif itself. Thus, with no loss of generality, we will assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq279_HTML.gif is connected.

If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq280_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq281_HTML.gif are http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq282_HTML.gif -manifolds of class http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq283_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq284_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq285_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq286_HTML.gif are http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq287_HTML.gif diffeomorphic, they are also http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq288_HTML.gif diffeomorphic ([10, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq289_HTML.gif , page 57]). Thus, since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq290_HTML.gif is of class http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq291_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq292_HTML.gif it suffices to find a bounded open subset http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq293_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq294_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq295_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq296_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq297_HTML.gif diffeomorphic to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq298_HTML.gif

In a first step, we find a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq299_HTML.gif function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq300_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq301_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq302_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq303_HTML.gif while http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq304_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq305_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq306_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq307_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq308_HTML.gif This can be done in various ways and even when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq309_HTML.gif However, since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq310_HTML.gif the most convenient argument is to rely on the fact that the signed distance function
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_Equ5_HTML.gif
(31)
is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq311_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq312_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq313_HTML.gif , and
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_Equ6_HTML.gif
(32)

is an open neighborhood of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq314_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq315_HTML.gif This is shown in Gilbarg and Trudinger [11, page 355] and also in Krantz and Parks [12]. Both proofs reveal that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq316_HTML.gif when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq317_HTML.gif that is, when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq318_HTML.gif (Without further assumptions, the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq319_HTML.gif regularity of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq320_HTML.gif breaks down when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq321_HTML.gif )

Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq322_HTML.gif be nondecreasing and such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq323_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq324_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq325_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq326_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq327_HTML.gif is given. Then, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq328_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq329_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq330_HTML.gif vanishes only on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq331_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq332_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq333_HTML.gif Furthermore, since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq334_HTML.gif on a neighborhood of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq335_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq336_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq337_HTML.gif on a neighborhood of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq338_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq339_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq340_HTML.gif remains http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq341_HTML.gif after being extended to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq342_HTML.gif by setting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq343_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq344_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq345_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq346_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq347_HTML.gif

This http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq348_HTML.gif satisfies all the required conditions except http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq349_HTML.gif Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq350_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq351_HTML.gif large enough, this can be achieved by replacing http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq352_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq353_HTML.gif Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq354_HTML.gif off http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq355_HTML.gif it follows from a classical theorem of Whitney [13, Theorem III] (with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq356_HTML.gif in that theorem) that there is a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq357_HTML.gif function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq358_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq359_HTML.gif of class http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq360_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq361_HTML.gif such that, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq362_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq363_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq364_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq365_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq366_HTML.gif

Evidently, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq367_HTML.gif does not vanish on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq368_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq369_HTML.gif has the same sign as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq370_HTML.gif off http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq371_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq372_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq373_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq374_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq375_HTML.gif Furthermore, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq376_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq377_HTML.gif so that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq378_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq379_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq380_HTML.gif Upon shrinking http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq381_HTML.gif we may assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq382_HTML.gif Also, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq383_HTML.gif For convenience, we summarize the relevant properties of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq384_HTML.gif below:

(i) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq385_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq386_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq387_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq388_HTML.gif off http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq389_HTML.gif ,

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq390_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq391_HTML.gif ,

(iii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq392_HTML.gif ,

(iv) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq393_HTML.gif ,

(v) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq394_HTML.gif

Choose http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq395_HTML.gif It follows from (v) that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq396_HTML.gif is compact and, from (iii) and (iv), that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq397_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq398_HTML.gif is small enough (argue by contradiction). Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq399_HTML.gif by (iii) and (iv) and since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq400_HTML.gif this implies http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq401_HTML.gif Thus, by (i) and (ii), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq402_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq403_HTML.gif submanifold of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq404_HTML.gif and the boundary of the open set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq405_HTML.gif In fact, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq406_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq407_HTML.gif -manifold of class http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq408_HTML.gif since, once again by (ii), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq409_HTML.gif lies on one side of its boundary.

We now proceed to show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq410_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq411_HTML.gif diffeomorphic to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq412_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq413_HTML.gif be such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq414_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq415_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq416_HTML.gif Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq417_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq418_HTML.gif by (ii), the function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq419_HTML.gif extended by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq420_HTML.gif outside http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq421_HTML.gif is a bounded http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq422_HTML.gif vector field on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq423_HTML.gif Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq424_HTML.gif the function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq425_HTML.gif defined by
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_Equ7_HTML.gif
(33)

is well defined and of class http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq426_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq427_HTML.gif is an orientation-preserving http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq428_HTML.gif diffeomorphism of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq429_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq430_HTML.gif We claim that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq431_HTML.gif produces the desired diffeomorphism from http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq432_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq433_HTML.gif

It follows at once from (3.3) that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq434_HTML.gif so that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq435_HTML.gif is decreasing along the flow lines and hence that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq436_HTML.gif maps http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq437_HTML.gif into itself for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq438_HTML.gif Also, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq439_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq440_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq441_HTML.gif so that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq442_HTML.gif by (iii). If now http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq443_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq444_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq445_HTML.gif is strictly decreasing for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq446_HTML.gif small enough. It follows that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq447_HTML.gif that is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq448_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq449_HTML.gif Altogether, this yields http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq450_HTML.gif

Suppose now that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq451_HTML.gif Then, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq452_HTML.gif and hence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq453_HTML.gif For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq454_HTML.gif small enough, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq455_HTML.gif and so http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq456_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq457_HTML.gif small enough. In fact, it is obvious that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq458_HTML.gif until http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq459_HTML.gif is large enough that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq460_HTML.gif But since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq461_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq462_HTML.gif is decreasing along the flow lines, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq463_HTML.gif implies http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq464_HTML.gif Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq465_HTML.gif this means that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq466_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq467_HTML.gif Call http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq468_HTML.gif the first (and, in fact, only, but this is unimportant) time when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq469_HTML.gif From the above, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq470_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq471_HTML.gif and hence for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq472_HTML.gif since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq473_HTML.gif Then, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq474_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq475_HTML.gif so that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq476_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq477_HTML.gif In particular, since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq478_HTML.gif and hence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq479_HTML.gif it follows that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq480_HTML.gif In other words, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq481_HTML.gif Thus, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq482_HTML.gif that is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq483_HTML.gif If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq484_HTML.gif (so that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq485_HTML.gif and hence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq486_HTML.gif ), this yields http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq487_HTML.gif On the other hand, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq488_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq489_HTML.gif Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq490_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq491_HTML.gif is strictly decreasing for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq492_HTML.gif near http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq493_HTML.gif and so http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq494_HTML.gif whence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq495_HTML.gif

The above shows that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq496_HTML.gif maps http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq497_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq498_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq499_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq500_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq501_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq502_HTML.gif That it actually maps http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq503_HTML.gif onto http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq504_HTML.gif follows from a Brouwer's degree argument: http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq505_HTML.gif is connected and no point of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq506_HTML.gif is in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq507_HTML.gif since, as just noted, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq508_HTML.gif Thus, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq509_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq510_HTML.gif is defined and independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq511_HTML.gif Now, choose http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq512_HTML.gif so that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq513_HTML.gif Then, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq514_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq515_HTML.gif and so http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq516_HTML.gif Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq517_HTML.gif is one to one and orientation-preserving, it follows that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq518_HTML.gif and so http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq519_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq520_HTML.gif Thus, there is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq521_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq522_HTML.gif which proves the claimed surjectivity.

At this stage, we have shown that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq523_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq524_HTML.gif diffeomorphism of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq525_HTML.gif mapping http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq526_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq527_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq528_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq529_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq530_HTML.gif into and onto http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq531_HTML.gif It is straightforward to check that such a diffeomorphism also maps http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq532_HTML.gif onto http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq533_HTML.gif (approximate http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq534_HTML.gif by a sequence from http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq535_HTML.gif ) and hence it is a boundary-preserving diffeomorphism of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq536_HTML.gif onto http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq537_HTML.gif This completes the proof of Lemma 1.3.

Remark 3.1.

The http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq538_HTML.gif diffeomorphism http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq539_HTML.gif above is induced by a diffeomorphism of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq540_HTML.gif but this does not mean that the same thing is true of the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq541_HTML.gif diffeomorphism of Lemma 1.3.

4. Systems

Suppose now that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq542_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq543_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq544_HTML.gif , is a system of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq545_HTML.gif differential operators on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq546_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq547_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq548_HTML.gif be two sets of Douglis-Nirenberg numbers, so that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq549_HTML.gif that have been normalized so that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq550_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq551_HTML.gif

It is well known that since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq552_HTML.gif proper ellipticity implies http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq553_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq554_HTML.gif We assume that a system http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq555_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq556_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq557_HTML.gif of boundary differential operators is given, with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq558_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq559_HTML.gif

Let
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_Equ8_HTML.gif
(41)

and call http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq560_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq561_HTML.gif the (complex valued) coefficients of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq562_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq563_HTML.gif respectively. Given an integer http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq564_HTML.gif introduce the following hypotheses (generalizing those for a single equation in the Introduction).

(H1; ) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq566_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq567_HTML.gif -submanifold of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq568_HTML.gif

(H2; ) The coefficients http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq570_HTML.gif are in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq571_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq572_HTML.gif and in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq573_HTML.gif otherwise.

(H3; ) The coefficients http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq575_HTML.gif are in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq576_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq577_HTML.gif and in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq578_HTML.gif otherwise.

For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq579_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq580_HTML.gif define
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_Equ9_HTML.gif
(42)

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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq581_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq582_HTML.gif ) and it is straightforward to check that the proof of Theorem 1.2 carries over to this case if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq583_HTML.gif If so, Corollary 2.1 is also valid, with a similar proof and an obvious modification of the function spaces.

Remark 4.1.

If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq584_HTML.gif there is no boundary condition (in particular, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq585_HTML.gif , and (H3; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq586_HTML.gif ) is vacuous) and the system http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq587_HTML.gif can be solved explicitly for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq588_HTML.gif in terms of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq589_HTML.gif and its derivatives. This is explained in [14, page 506]. If so, the smoothness of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq590_HTML.gif (i.e., (H1; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq591_HTML.gif ) is irrelevant, and Theorem 1.2 is trivially true regardless of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq592_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq593_HTML.gif is an isomorphism). A special case when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq594_HTML.gif arises if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq595_HTML.gif (in particular, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq596_HTML.gif ), for then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq597_HTML.gif from the conditions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq598_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq599_HTML.gif

From the above, Theorem 1.2 may only fail if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq600_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq601_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq602_HTML.gif . (The author was recently informed by H. Koch [16] that he could prove Lemma 1.3 when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq603_HTML.gif so that Theorem 1.2 remains true in this case as well.)

Authors’ Affiliations

(1)
Department of Mathematics, University of Pittsburgh

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Copyright

© Patrick J. Rabier 2009

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