Open Access

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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq3_HTML.gif . Here, https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq7_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq8_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq9_HTML.gif denote a differential operator on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq10_HTML.gif of order https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq11_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq12_HTML.gif with complex coefficients,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_Equ1_HTML.gif
(11)
Next, let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq14_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq15_HTML.gif of order https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq16_HTML.gif also with complex coefficients,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_Equ2_HTML.gif
(12)

With https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq17_HTML.gif and https://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; ) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq20_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq21_HTML.gif -submanifold of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq22_HTML.gif (i.e., https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq23_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq24_HTML.gif submanifold of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq25_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq26_HTML.gif lies on one side of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq27_HTML.gif );

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

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

Then, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq38_HTML.gif , the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq39_HTML.gif maps continuously https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq40_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq41_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq42_HTML.gif maps continuously https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq43_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq44_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq45_HTML.gif
https://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; https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq46_HTML.gif ), (H2; https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq47_HTML.gif ), and (H3; https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq48_HTML.gif ) hold for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq49_HTML.gif Then,

(i)if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq50_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq51_HTML.gif the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq52_HTML.gif is Fredholm if and only if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq53_HTML.gif is uniformly elliptic in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq54_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq56_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq57_HTML.gif is Fredholm for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq58_HTML.gif and some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq59_HTML.gif (and hence for every such https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq60_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq62_HTML.gif are independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq63_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq65_HTML.gif

Note that https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq69_HTML.gif replaced by https://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] ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq71_HTML.gif -independence for https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq73_HTML.gif under additional technical conditions).

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq74_HTML.gif is invertible for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq75_HTML.gif and every https://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, https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq79_HTML.gif from the case https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq81_HTML.gif regularity of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq82_HTML.gif since Theorem 1.1 with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq83_HTML.gif can only be used if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq84_HTML.gif is https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq86_HTML.gif is a bounded open subset of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq87_HTML.gif and that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq88_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq89_HTML.gif -submanifold of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq90_HTML.gif of class https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq91_HTML.gif with https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq93_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq94_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq95_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq96_HTML.gif -submanifold of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq97_HTML.gif of class https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq98_HTML.gif (even https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq99_HTML.gif ) and that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq100_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq101_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq102_HTML.gif diffeomorphic (as https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq104_HTML.gif -manifold https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq105_HTML.gif of class https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq106_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq107_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq108_HTML.gif diffeomorphic to a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq109_HTML.gif -manifold https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq110_HTML.gif of class https://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 https://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 https://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 https://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; https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq115_HTML.gif ), (H2; https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq116_HTML.gif ), and (H3; https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq118_HTML.gif is not connected, the system https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq121_HTML.gif -independence of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq122_HTML.gif (recall https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq123_HTML.gif ) follows from that of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq125_HTML.gif and https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq129_HTML.gif are in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq130_HTML.gif and all the coefficients https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq131_HTML.gif are in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq132_HTML.gif which is henceforth assumed.

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

The pull-back https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq147_HTML.gif is a linear isomorphism of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq148_HTML.gif onto https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq149_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq150_HTML.gif and of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq151_HTML.gif onto https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq152_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq153_HTML.gif Meanwhile, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq154_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq155_HTML.gif is a differential operator of order https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq156_HTML.gif with coefficients https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq157_HTML.gif of class https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq158_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq159_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq160_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq161_HTML.gif is a differential operator of order https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq162_HTML.gif with coefficients https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq163_HTML.gif of class https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq164_HTML.gif on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq166_HTML.gif )
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_Equ4_HTML.gif
(21)

has the form https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq167_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq168_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq169_HTML.gif are isomorphisms. As a result, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq171_HTML.gif Since the coefficients of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq172_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq173_HTML.gif and of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq174_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq176_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq177_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq178_HTML.gif by https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq180_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq181_HTML.gif submanifold of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq182_HTML.gif (but the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq183_HTML.gif are still https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq184_HTML.gif and the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq185_HTML.gif still https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq186_HTML.gif ).

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

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq203_HTML.gif be fixed. The corresponding operators https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq204_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq205_HTML.gif are arbitrarily norm-close to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq206_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq208_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq209_HTML.gif are Fredholm with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq210_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq211_HTML.gif But since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq212_HTML.gif is now https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq213_HTML.gif and the coefficients https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq214_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq215_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq216_HTML.gif the hypotheses (H1; https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq217_HTML.gif ), (H2; https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq218_HTML.gif ), and (H3; https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq219_HTML.gif ) are satisfied by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq220_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq221_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq222_HTML.gif and any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq223_HTML.gif Thus, https://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 https://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; https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq226_HTML.gif ), (H2; https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq227_HTML.gif ), and (H3; https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq228_HTML.gif ) hold, that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq229_HTML.gif is uniformly elliptic in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq230_HTML.gif , and that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq231_HTML.gif satisfies the LS condition. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq232_HTML.gif If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq233_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq234_HTML.gif , then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq236_HTML.gif we assume https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq237_HTML.gif Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq238_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq240_HTML.gif denote a (finite-dimensional) complement of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq241_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq242_HTML.gif Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq243_HTML.gif is dense in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq244_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq245_HTML.gif is closed, we may assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq246_HTML.gif If not, approximate a basis of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq247_HTML.gif by elements of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq249_HTML.gif (of dimension https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq250_HTML.gif ) intersects https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq251_HTML.gif only at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq252_HTML.gif (by the closedness of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq253_HTML.gif ). Thus, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq254_HTML.gif may be replaced by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq255_HTML.gif as a complement of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq256_HTML.gif .

Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq257_HTML.gif and https://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, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq259_HTML.gif because https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq260_HTML.gif is a complement of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq261_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq262_HTML.gif This shows that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq263_HTML.gif is also a complement of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq264_HTML.gif

Therefore, since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq265_HTML.gif there is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq266_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq267_HTML.gif This yields https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq268_HTML.gif whence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq269_HTML.gif and so https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq270_HTML.gif This means that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq271_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq272_HTML.gif Thus, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq273_HTML.gif that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq274_HTML.gif Since https://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 https://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, https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq279_HTML.gif is connected.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq280_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq281_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq282_HTML.gif -manifolds of class https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq283_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq284_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq285_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq286_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq287_HTML.gif diffeomorphic, they are also https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq288_HTML.gif diffeomorphic ([10, Theorem https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq289_HTML.gif , page 57]). Thus, since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq290_HTML.gif is of class https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq291_HTML.gif with https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq293_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq294_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq295_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq296_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq297_HTML.gif diffeomorphic to https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq299_HTML.gif function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq300_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq301_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq302_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq303_HTML.gif while https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq304_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq305_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq306_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq307_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq309_HTML.gif However, since https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_Equ5_HTML.gif
(31)
is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq311_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq312_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq313_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_Equ6_HTML.gif
(32)

is an open neighborhood of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq314_HTML.gif in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq316_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq317_HTML.gif that is, when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq318_HTML.gif (Without further assumptions, the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq319_HTML.gif regularity of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq320_HTML.gif breaks down when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq321_HTML.gif )

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

This https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq348_HTML.gif satisfies all the required conditions except https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq349_HTML.gif Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq350_HTML.gif for https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq352_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq353_HTML.gif Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq354_HTML.gif off https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq357_HTML.gif function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq358_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq359_HTML.gif of class https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq360_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq361_HTML.gif such that, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq362_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq363_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq364_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq365_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq366_HTML.gif

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

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

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

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

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

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

Choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq395_HTML.gif It follows from (v) that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq397_HTML.gif if https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq399_HTML.gif by (iii) and (iv) and since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq400_HTML.gif this implies https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq401_HTML.gif Thus, by (i) and (ii), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq402_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq403_HTML.gif submanifold of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq405_HTML.gif In fact, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq406_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq407_HTML.gif -manifold of class https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq408_HTML.gif since, once again by (ii), https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq410_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq411_HTML.gif diffeomorphic to https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq413_HTML.gif be such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq414_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq415_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq416_HTML.gif Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq417_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq418_HTML.gif by (ii), the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq419_HTML.gif extended by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq420_HTML.gif outside https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq421_HTML.gif is a bounded https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq422_HTML.gif vector field on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq423_HTML.gif Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq424_HTML.gif the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq425_HTML.gif defined by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq426_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq427_HTML.gif is an orientation-preserving https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq428_HTML.gif diffeomorphism of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq429_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq430_HTML.gif We claim that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq431_HTML.gif produces the desired diffeomorphism from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq432_HTML.gif to https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq434_HTML.gif so that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq436_HTML.gif maps https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq437_HTML.gif into itself for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq438_HTML.gif Also, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq439_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq440_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq441_HTML.gif so that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq442_HTML.gif by (iii). If now https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq443_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq444_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq445_HTML.gif is strictly decreasing for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq446_HTML.gif small enough. It follows that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq447_HTML.gif that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq448_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq449_HTML.gif Altogether, this yields https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq450_HTML.gif

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

The above shows that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq496_HTML.gif maps https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq497_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq498_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq499_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq500_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq501_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq502_HTML.gif That it actually maps https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq503_HTML.gif onto https://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: https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq505_HTML.gif is connected and no point of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq506_HTML.gif is in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq507_HTML.gif since, as just noted, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq508_HTML.gif Thus, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq509_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq510_HTML.gif is defined and independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq511_HTML.gif Now, choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq512_HTML.gif so that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq513_HTML.gif Then, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq514_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq515_HTML.gif and so https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq516_HTML.gif Since https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq518_HTML.gif and so https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq519_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq520_HTML.gif Thus, there is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq521_HTML.gif such that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq523_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq524_HTML.gif diffeomorphism of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq525_HTML.gif mapping https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq526_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq527_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq528_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq529_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq530_HTML.gif into and onto https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq532_HTML.gif onto https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq533_HTML.gif (approximate https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq534_HTML.gif by a sequence from https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq536_HTML.gif onto https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq538_HTML.gif diffeomorphism https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq542_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq543_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq544_HTML.gif , is a system of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq545_HTML.gif differential operators on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq547_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq549_HTML.gif that have been normalized so that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq550_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq551_HTML.gif

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

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

and call https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq560_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq561_HTML.gif the (complex valued) coefficients of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq562_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq563_HTML.gif respectively. Given an integer https://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; ) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq566_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq567_HTML.gif -submanifold of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq568_HTML.gif

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

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

For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq579_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq580_HTML.gif define
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq581_HTML.gif and https://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 https://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 https://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, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq585_HTML.gif , and (H3; https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq586_HTML.gif ) is vacuous) and the system https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq587_HTML.gif can be solved explicitly for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq588_HTML.gif in terms of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq590_HTML.gif (i.e., (H1; https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq592_HTML.gif ( https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq594_HTML.gif arises if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq595_HTML.gif (in particular, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq596_HTML.gif ), for then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq597_HTML.gif from the conditions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq598_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq600_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F637243/MediaObjects/13661_2009_Article_868_IEq601_HTML.gif , and https://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 https://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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