USA-Chile Workshop on Nonlinear Analysis, Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 1–11 http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)

An estimate on the relative Morse index for strongly indefinite functionals ^{∗}

A. Abbondandolo, P. Felmer, & J. Molina

Abstract

We extend the Benci and Rabinowitz linking theorem to strongly indefinite functionals satisfying the Palais-Smale condition. More precisely, we show an upper estimate for a relative Morse index of critical points.

1 Introduction

Let E be a real Hilbert space, with scalar product (·, ·). On this space. we consider the functional

f(u) = \frac{1}{2}(Lu, u) + h(u) ,  (1)

where L is an invertible self-adjoint operator and h ∈ C^{2}(E) with h′ is compact. In particular, we are interested in the strongly indefinite case, that is when both the positive and the negative eigenspaces of L are infinite dimensional. It is well known that critical points of such functionals always have infinite Morse index.

For a closed subspace of V ⊂ E, we denote by P_{V} the orthogonal projection onto V and by V^{⊥} the orthogonal complement of V. Following [1] and [2], we say that the closed subspaces V, W of E are commensurable if P_{V^{⊥}}P_{W} and P_{W^{⊥}}P_{V} are compact operators.

If V and W are commensurable, the relative dimension of W with respect to V is defined as

dim_{V} W = dim W ∩ V^{⊥} − dim W^{⊥} ∩ V.

Commensurability guarantees that both terms in the above formula are finite. It can be proved that if two self-adjoint Fredholm operators differ by a compact operator, then their negative (resp. positive) eigenspaces are commensurable.

Denote by E^{+} and E^{−} the positive and the negative eigenspaces of L. Since f′′(x) = L + h′′(x) and h′′(x) is compact, the relative Morse index of a critical point x of functional f with respect to the splitting E = E^{+} ⊕E^{−} can be defined as the integer

∗Mathematics Subject Classifications: 58E05.

Key words: Critical point, Morse index.

\begin{matrix}c \\ ◯ \end{matrix}2000 Southwest Texas State University.

Published January 8, 2001.

Partially supported by FONDAP de Matemáticas Aplicadas.

J.M. was also supported by FONDECYT grant 1990349 .

m_{(E^{+},E^{−})}(x) = dim_{E^{−}}[ negative eigenspace of f′′(x) ].

For degenerate critical points, a relative large Morse index can be defined as well:

m_{(E^{+},E^{−})}^{∗}(x) = m_{(E^{+},E^{−})}(x) + dim ker f′′(x).

In [3] it is shown that this definition of the Morse index coincides with a definition given by Chang, Liu and Liu [7] which uses finite dimensional projections. Chang’s definition fits very well in the framework of Galerkin reductions. Assuming the (P S^{∗}) condition, Galerkin reductions and the equivalence of the two indices are used in [3] to prove Morse index estimates for critical points coming from Benci and Rabinowitz’s generalized Mountain Pass theorem.

Here we give another proof of the upper index estimate without using finite dimensional reductions. The advantage of this approach is that we have only to assume the usual (P S) condition, and not the stronger (P S^{∗}) condition. Here is our result.

Theorem 1.1 Let E = W^{+} ⊕ W^{−} be an orthogonal decomposition of E, where W^{+} is commensurable with E^{+} and W^{−} is commensurable with E^{−}. Let e ∈ ∂B_{1} ∩ W^{+} and set

S = ∂B_{ρ} ∩ W^{+}, Q = B_{s} ∩ W^{−} ⊕ [0, r]e,

where r > ρ > 0 and s > 0. Denote by ∂Q the boundary of Q in W^{−} ⊕ ℝe. Assume that there exist numbers α < β such that

sup_{∂Q} f < α < inf_{S} f, sup_{Q} f < β,

and that f satisfies the (P S)_{c} condition for every c ∈ [α, β].

Then f has a critical point x such that α ≤ f(x) ≤ β. Moreover, if f ∈ C^{2}(E), the following index estimate hold

m_{(E^{+},E^{−})}(x) ≤ dim_{E^{−}} W^{−} + 1.

This estimate generalizes the usual index estimate of finite dimensional minmax critical points, proved by Lazer and Solimini [9], Solimini [13] and Benci [5].

2 The relative dimension

Let E be a real Hilbert space, with scalar product (·, ·). We deal with functionals of the form

f(u) = \frac{1}{2}(Lu, u) + h(u),

where L is an invertible self-adjoint operator, h ∈ C^{2}(E) and h′ is compact. Moreover, denote by E^{+} and E^{−} the positive and the negative eigenspaces of L. For a closed subspace V ⊂ E, denote by P_{V} the orthogonal projection onto V and by V^{⊥} the orthogonal complement of V.

Definition 2.1 Two closed subspaces V and W of E are called commensurable if the operator P_{V} − P_{W} is compact. In this case the relative dimension of W with respect to V is the integer

dim_{V} W := dim W ∩ V^{⊥} − dim W^{⊥} ∩ V.

The identities

P_{V} − P_{W} = P_{V} P_{W^{⊥}} + P_{V} P_{W} − P_{W} = (P_{W^{⊥}}P_{V})^{∗} − P_{V^{⊥}}P_{W} ,
P_{V^{⊥}}P_{W} = (P_{W} − P_{V})P_{W} , P_{W^{⊥}}P_{V} = (P_{V} − P_{W})P_{V} ,

show that V and W are commensurable if and only if both P_{W^{⊥}}P_{V} and P_{V^{⊥}}P_{W} are compact. Hence the spaces W ∩ V^{⊥} and W^{⊥} ∩ V in the above definition are finite dimensional because they are the spaces of fixed points of the compact operators P_{V^{⊥}}P_{W} and P_{W^{⊥}}P_{V}, respectively.

Easy examples of pairs of commensurable subspaces are built by adding or removing a finite number of dimensions: V ⊕W_{r} is commensurable with V if W_{r} has dimension r, and dim_{V} (V ⊕W_{r}) = r. If W^{s} is a s-codimensional subspace of V, then it is commensurable with V and dim_{V} W^{s} = −s. Clearly the definition of commensurability is more general than this: for example the intersection of two commensurable spaces could be {0}. Here are some useful facts which follow directly from the definitions. For V, W, Z commensurable subspaces,

(i) dim_{W} V = − dim_{V} W;

(ii) dim_{Z} V = dim_{W} V + dim_{Z} W;

(iii) V^{⊥} and W^{⊥} are commensurable and dim_{V^{⊥}} W^{⊥} = − dim_{V} W.

Formula (ii) with Z = {0} implies that dim_{W} V = dim V − dim W when V and W are finite dimensional.

Remark 2.1 Commensurability is an equivalence relation. The notion of commensurability is stable with respect to operators of the form identity + compact

The proof of the following result it is found in [3] [Theorem 2.6]

Proposition 2.1 Let T be a self-adjoint Fredholm operator. If K is a selfadjoint compact operator, the positive (negative) eigenspaces of T and T + K are commensurable.

Vice versa, if T is an invertible self-adjoint operator with positive (negative) eigenspace E^{+} (E^{−}) and E = W^{+} ⊕ W^{−}, where W^{+} (W^{−}) is commensurable with E^{+} (E^{−}), then there exists an invertible self-adjoint operator M whose positive (negative) eigenspace is W^{+} (W^{−}) and such that M − T is compact.

From the above proposition we get the following

Remark 2.2 Let X, Y two closed commensurables subspaces of the Hilbert space H. If dim_{Y} X > 0 then there exists T : H → H with T = I + K invertible such that T (X) = Y ⊕ Y_{r} with Y_{r} ≠ 0 and dim Y_{r} < ∞.

Definition 2.2 Let x be a critical point of the functional f. Assume that f is twice differentiable in x. Let E = V^{+} ⊕ V^{−} ⊕ V^{0} be the decomposition of E into the positive, the negative and the null eigenspaces of f′′(x) = L + K, where K is a compact self-adjoint operator. The relative Morse index of x with respect to the splitting E^{+} ⊕ E^{−} of H is the integer

m_{(E^{+},E^{−})}(x) = m_{(E^{+},E^{−})}(x; f) = dim_{E^{−}} V^{−}.

The following results are from [3], Theorem 1.5 and Proposition 1.3 respectively.

Proposition 2.2 Let (K_{n}) be a sequence of self-adjoint compact operators which converges to K in the operators’ norm. Moreover, let (f_{n}) be a C^{2} sequence in x and suppose that f′′_{n}(x) = L + K_{n} and f′′ = L + K. Then

m_{(E^{+},E^{−})}(x; f) ≤ lim inf_{n→∞} m_{(E^{+},E^{−})}(x; f_{n}).  (2)

3 The non degenerate case

In this section we will see that in order to prove Theorem 1.1, is suffices to show only the case of non degenerate critical points.

Theorem 3.1 Under the hypothesis of Theorem 1.1 and assuming that all critical points at level c are non degenerate there exits a critical point u_{0} such that f(u_{0}) = c and m_{(W^{+},W^{−})}(u_{0}) ≤ 1.

Definition 3.1 Let U(x) be an open neighborhood of x ∈ E. We say that f ∈ C^{2}(U(x)) is a Morse function on U(x), if the following condition holds.

(i) f(x) satisfies (PS) on U(x).

(ii) f has only non degenerate critical points on U(x).

Remark 3.1 Let x be a degenerate critical point of f ∈ C^{2}(U(x)). Assume that, f satisfies the (PS) condition on U(x) and f′′(x) = T + K where T is an invertible self-adjoint operator and K is self-adjoint compact operator.

Then from the celebrated Marino-Prodi’s Theorem, there exists a Morse sequence (f_{n}) on U(x) such that

f_{n} = f on X − B(x, \frac{1}{n}) for all n
f_{n} → f in C^{2}(U(x)) .

Remark 3.2 Observe that assuming Theorem 3.1, the prove of the Theorem 1.1 is easy.

In fact, let u_{0} be a degenerate critical point of f. So, there exists a sequence {u_{n}} such that u_{n} → u_{0} and f′(u_{n}) = 0. On the other hand, it is easy to see that f satisfies all assumptions of Remark 3.1. So, let (f_{n}) be a strong Morse sequence such that

f_{n} = f on X − B(u_{0}, \frac{1}{n^{2}} ) for all n
f_{n} → f in C^{2}(U(u_{0})).

and hence we can assume f′_{n}(u_{n}) = 0. Moreover note that for all n big, f_{n} satisfies the hypothesis of Theorem 1.1, and therefore we can assume that u_{n} is a min-max for f_{n} with f_{n}(u_{n}) =: c_{n}. Observe that the u_{n} are non degenerate critical points of f_{n}, so from Theorem 3.1 we obtain that

m_{(W^{+},W^{−})}(u_{n}, f_{n}) ≤ 1

for all n big. Now, by Proposition 2.2 it is follows that

m_{(W^{+},W^{−})}(u_{0}, f_{n}) ≤ 1

and hence, finally, we obtain that m_{(W^{+},W^{−})}(u_{0}, f) ≤ 1.

4 Proof of the Theorem 1.1

From Proposition 2.1 changing L by another invertible self-adjoint operator, and hence changing h by another map with compact gradient, we can assume in Theorem 1.1 that W^{+} = E^{+} and W^{−} = E^{−}.

Let Γ = {g ∈ C([0, 1] × E, E) : g satisfies Γ_{1}, Γ_{2}} where

(Γ_{1}) g(u) = u, for u ∈ ∂Q;

(Γ_{2}) γ(u) = s(u)

exp{θ(u)L}u+K(u) where θ : E → ℝ is a Lipschitz function, K : E → E is a compact and Lipschitz function and s : E →]0, 1] is a Lipschitz function with s(E) > s_{0} > 0;

As in Theorem 5.3 of [11], is easy to prove that

c = inf_{g∈Γ} sup_{u∈Q} f(g(u))

is a critical value of f.

Because of the term of s(u) in (Γ_{2}) we have to prove that c ≥ α. To do that, is suffices to show that for all g ∈ Γ, g(Q) ∩ S_{ρ} ≠ ∅.

For λ ∈ ℝ and u ∈ E^{−}, we define:

Ψ(u) = Ψ(λe + u^{−}) = ∥P_{E^{+}}γ(u)∥e + \frac{1}{s(u)}e^{−θ(u)L}P_{E^{−}}γ(u).

From the fact that e^{θL} commutes with P_{E^{−}}, we get that

Ψ(u) = ∥P_{E^{+}}γ(λe + u^{−})∥e + u^{−} + \frac{1}{s(λe + u^{−})}e^{−θ(λe+u^{−})L}P_{E^{−}}K(λe + u^{−}).

On the other hand, using the fact that \frac{1}{|s|} ≤ \frac{1}{s_{0}} < +∞, θ is bounded and K is a compact map, we obtain that Ψ is a compact perturbation of the identity.

Moreover, if u ∈ ∂Q, by (Γ_{1}) and (Γ_{2}) we get that Ψ(u) = u. Hence, Ψ(u) ≠ ρe if u ∈ ∂Q and

deg(Ψ, Q, ρe) = deg(id, Q, ρe) = 1.

Therefore there exists ū ∈ Q such that Ψ(ū) = ρe, that is, γ(ū) ∈ S_{ρ}. The remaining part of the proof it follows from Theorem 3.1 and Remark 3.2.

5 Results used for proving Theorem 3.1

Let u_{0} be a non degenerate critical point of f with f(u_{0}) = c and assume that E = V^{−} ⊕ V^{+} where V^{−} (respectively V^{+}) is the negative eigenspace (positive eigenspace) of f′′(u_{0}). Also we write P_{−} (respectively P_{+}) the projection of E on V^{−} (V^{+}). Moreover, if u ∈ E, by u_{−} we means P_{−}(u) (respectively for u_{+}).

Let B_{−} = B(0, r_{1})∩V_{−} and B_{+} = B(0, r_{2})∩V^{+}. For r_{1} > 0 and r_{2} > 0 small enough, by Morse Lemma, there exists a local C^{1}-isomorphism Ψ : 2B_{−} ⊕B_{+} → U where Ψ(0) = u_{0} and U = Ψ(2B_{−} ⊕ B_{+}), such that

f(Ψ(z_{+} + z_{−})) = c + ||z_{+}||^{2} − ||z_{−}||^{2}  (3)

Now for r_{1} and r_{2} such that r_{2}^{2} − 4r_{1}^{2} > 0, we define Φ : E → E as follows

Φ(u) = \{ \begin{matrix}u & if u ∉ U \\ Ψ(φ(\frac{||Ψ^{−1}(u)_{−}||}{r_{1}} − 1)Ψ^{−1}(u)_{+} + Ψ^{−1}(u)_{−}) & if u ∈ U \end{matrix}

where φ : ℝ → [0, 1] is a Lipschitz function defined by

φ(s) = \{ \begin{matrix}0 & if s ≤ 0 \\ 1 & if s ≥ 1 \end{matrix}

Note that Φ is continuous on H −Ψ(2B_{−} ⊕∂B_{+}); and Φ is Lipschitz on f^{−1}(]− ∞, c + α_{1}]) with α_{1} as in Lemma 5.1 below.

The following Lemma is similar to Lemma 3.1 in [9], except for (iv).

Lemma 5.1 Let g ∈ Γ, 0 < α_{1} < r_{2}^{2} − 4r_{1}^{2}, c = f(u_{0}) and U_{1} = Ψ(B_{−} ⊕ B_{+})

(i) If u ∈ U − int(U_{1}) then f(Φ(u)) ≤ f(u);

(ii) If u ∈ ∂U_{1} and f(u) ≤ c + α_{1} then Φ(u) = Ψ({Ψ^{−1}(u)}_{−}) and Φ(u) ∈ Ψ(∂B_{−});

(iii) Φ(H − int(U_{1})) ⊂ H − int(U_{1});

(iv) Suppose that f(g(Q)) < c + α_{1}, and f(∂Q) ≤ α < c. Moreover assume that A ≠ ∅ where A = g^{−1}(int(U_{1})) ∩ Q. Then Φ(g(∂A)) ∈ Ψ(∂B_{−}(0, r_{1})) .

Proof.

(i) It is follows easily from (3).

(ii) There result that Ψ^{−1}(u) ∈ (∂B_{−} ⊕ B_{+}) ∪ (B_{−} ⊕ ∂B_{+}). But, Ψ^{−1}(u) ∈ B_{−} ⊕ ∂B_{+} it is not possible otherwise

f(u) = c + ||Ψ^{−1}(u)_{+}||^{2} − ||Ψ^{−1}(u)_{−}||^{2} > c + α

Thus, Ψ^{−1}(u) ∈ ∂B_{−} ⊕ B_{+} and therefore φ(\frac{||Ψ^{−1}(u)_{−}||}{r_{1}} − 1) = 0 and Φ(u) = Ψ[Ψ^{−1}(u)_{−}] ∈ Ψ(∂B_{−}).

(iii) It is follows from definition.

(iv) It is easy to see that g(u) ∈ ∂U_{1} when u ∈ ∂A, and

f(g(∂A)) ≤ c + α_{1}

On the other hand, we can assume that g^{−1}(int(U_{1})) ∩ ∂Q = ∅.

Otherwise, if z is such that Ψ(z) ∈ ∂Q ∩ (int(U_{1})), then f(Ψ(z)) ≤ α_{1}. But the last is not possible, if we take r_{1} in the definition of Ψ, such that c − α_{1} > r_{1}^{2}.

The two following Lemmas will be important in the next section; the first one is known as Mac Shane Lemma and its proof can be found in [8]. The proof of the second one can be found in [10].

Lemma 5.2 Let X be a metric space, Y ⊂ X and α > 0. If f : Y → ℝ is a Lipschitz function with constant α, then f can be extended to X as a Lipschitz function with the same constant α.

Lemma 5.3 Let H be a Hilbert space and let f : H → ℝ be a uniformly continuous function. Then f can be approximate uniformly by C^{1} functions.

We also need the following result

Lemma 5.4 Let E and F two separable Banach spaces. Moreover let h be a C^{1} Fredholm map of index 0 of E into F, such that w ∉ h(E) for some w ∈ F. Then h is not locally surjective.

Proof. Let us assume by contradiction that h is locally surjective. Then for y_{1} = h(x_{1}) we can take y near to y_{1} such that there exists x ∈ h^{−1}(y). Hence, from Sard-Smale Theorem, h′(x) is surjective and we get that h is locally invertible in x. But this would imply that h is globally invertible, which contradicts the fact that w ∉ h(E).

6 Proof of Theorem 3.1

We observe that there only finitely many critical points in K_{c} since f satisfies (PS) and all critical points are non degenerate.

Remember that, cf [11], given ϵ > 0 and W any neighborhood of the set of critical points at level c, there exist ϵ_{1} ∈]0, ϵ[ and η : [0, 1] × E → E continuous such that

(i) η(0, u) = u for all u.

(ii) η(t, u) = u for all t ∈ [0, 1] if f(u) ∉ [c − ϵ_{1}, c + ϵ_{1}]).

(iii) η(1, f^{c+ϵ_{1}} − W) ⊂ f^{c−ϵ_{1}}

We first treat the case when K_{c} is the singleton {u_{0}}. The general case is obtained by induction.

Let us assume that dim_{W_{1}} V^{−} > 0 (here W_{1} = W^{−} ⊕ {λe : λ ∈ ℝ}). It is enough to prove the existence of an open set Ũ containing u_{0} so that for all ϵ > 0 small enough there exists γ ∈ Γ (see Section 4 for definition of Γ) such that

(F)  sup_{u∈Q} f(γ(u)) < c + ϵ and γ(Q) ∩ Ũ = ∅

This together with a deformation argument lead to a contradiction:

In fact, because c ≥ α and sup_{∂Q} f < α, there exists ϵ_{1}, with 0 < ϵ_{1} < ϵ such that

max_{u∈∂Q} f(u) < c − 2ϵ_{1}

Let g ∈ Γ such that g satisfies (F). If we define g_{1}(u) = η(1, g(u)), we have that for u ∈ ∂Q, g_{1}(u) = η(1, g(u)) = g(u) = u, so g_{1} ∈ Γ (see Theorem 5.29 of [11]).

Now, from (F) we get that g_{1}(Q) ⊂ f^{−1}(] − ∞, c − ϵ]) and hence sup_{u∈Q} f(g_{1}(u)) < c − ϵ < c which is a contradiction. Construction of γ: Let g ∈ Γ such that sup_{u∈Q} f(g(u)) < c+ϵ_{1} where ϵ_{1} < α_{1} with α_{1} as in Lemma 5.1. If we write A = g^{−1}(int(U_{1}))∩Q, we can assume that A ≠ ∅, since otherwise g(Q) ∩ U_{1} = ∅ and we get the condition (F). Moreover, as in the proof of Lemma 5.1 (iv), we can assume that g^{−1}(int(U_{1})) ∩ ∂Q = ∅.

From Lemma 5.1, Φ(g(∂A)) ∈ Ψ(∂B_{−}(0, r_{1})). Actually, for u ∈ ∂A there result that Φ(g(u)) = Ψ[Ψ^{−1}(g(u))_{−}].

On the other hand, it is easy to see that (cf. [12])

Ψ = (I − K)|f′′(u_{0})|^{− \frac{1}{2}} Ψ^{−1} = |f′′(u_{0})|^{\frac{1}{2}} (I + K_{1})

where K and K_{1} are compact.

From now on K_{i} always will denote a compact map. We claim that (Ψ^{−1} ∘ g)|_{∂A} has a smooth extension on cl(A). In fact for u ∈ ∂A

Ψ^{−1}(g(u)) = |f′′(u_{0})|^{\frac{1}{2}} (I + K_{1})(g(u))  (4)
= |f′′(u_{0})|^{\frac{1}{2}} s(u) exp\{Θ(u)L\}u + K(u).  (5)

For all ϵ > 0 small enough there exists a compact map K_{ϵ} with finite rank defined on ∂A such that ||K_{ϵ}(u) − K(u)|| < ϵ. By using Lemma 5.2 and 5.3 there exists an smooth extension K_{4} of K_{ϵ}. In the same way, let θ_{1} be an extension of class C^{1}(cl(A)) of θ_{|∂A}. Now, the extension to cl(A) of (Ψ^{−1} ∘g)|_{∂A} is Ψ_{1}(u) = |f′′(u_{0})|^{\frac{1}{2}} s(u) exp\{Θ_{1}(u)L\}u + K_{4}(u).

Because,

exp{Θ_{1}(u)L} − exp{Θ_{1}(u)[L + h′′(u_{0})]}eqno(C)

is compact, we get that Ψ_{1}(u) = |f′′(u_{0})|^{\frac{1}{2}} s(u) exp\{Θ(u)[L + h′′(u_{0})]\}u + K_{5}

On the other hand, from dim_{W_{1}} V^{−} > 0 and using Remark 2.2, we get for u ∈ cl(A), that P_{−}(u) = u − K_{A}(u) where K_{A} is compact. Hence,

P_{−}(Ψ_{1})(u) = s(u)|f′′(u_{0})|^{\frac{1}{2}} exp\{Θ_{1}(u)[L + h′′(u_{0})]\}u + K_{6}(u).

Moreover, using again (C), we get that Ψ_{2} = P_{−}(Ψ_{1}) is Fredholm map of index 0 (Ψ_{2}|_{cl(A)} : cl(A) → V^{−}).

In fact, dΨ_{2}(u)(v) = s(u)|f′′(u_{0})|^{\frac{1}{2}} exp\{Θ_{1}(u)[L + h′′(u_{0})]\}\{I + K\}(v).

Now let T be as in Remark 2.2. It follows from Lemma 5.4 that T ∘ Ψ_{2} is not locally surjective and hence Ψ_{2} it is not locally surjective either. That is, there exists δ ∈]\frac{1}{2}r_{1}, \frac{3}{4}r_{1}[ and z ∈ B(0, δ) ∩ V^{−} such that z ∉ Im(Ψ_{2}). Now, we can project Im(Ψ_{2}) ∩ B(0, δ) on ∂B(0, δ) from z. Thus, we can think that Ψ_{2} maps cl(A) into V^{−} − B(0, δ). Moreover, we note that Ψ_{2} is bounded on bounded sets.

Now, we define Ψ_{3}(u) = \frac{δ}{||Ψ_{2}(u)||}Ψ_{2}(u), that is,

Ψ_{3}(u) = \frac{δ}{||Ψ_{2}(u)||}|f′′(u_{0})|^{\frac{1}{2}} s(u) exp\{Θ_{1}(u)L\}u + K_{6}(u) .

Note that actually Ψ_{3}(cl(A)) ⊂ ∂B(0, δ) ∩ V^{−}.

Thus, we get that Ψ ∘ Ψ_{3}(u) = s_{1}(u) exp{Θ_{1}(u)L}u + K_{7}(u) where s_{1}(u) = \frac{δ}{||Ψ_{2}(u)||}s(u) ∈]s_{0}, 1[, with s_{0} > 0, and Ψ ∘ Ψ_{3}(cl(A)) ⊂ U − Ψ(B(0, δ)).

Finally, we define

γ(u) = \{ \begin{matrix}Φ(g(u)) & if u ∉ A \\ Ψ ∘ Ψ_{3}(u) & if u ∈ cl(A) \end{matrix}

Observe that actually our extension Ψ∘Ψ_{3} may differ from Φ∘g on some points of the boundary ∂A, but since Ψ ∘ Ψ_{3}(∂A) is outside Ψ(B(0, δ)), we can think of Φ ∘ g and Ψ ∘ Ψ_{3} as being the same on ∂A.

Note that the condition (F) holds for γ and B(u_{0}, δ). Suppose that u ∉ A, so we can assume that g(u) ∉ int(U_{1}) (and u ∈ Q). From Lemma 5.1, Φ(g(u)) ∈ H − int(U_{1}). Therefore, if u ∉ A, γ(u) ∉ Ψ(B(0, δ)). Assume now that u ∈ cl(A). By definition γ(u) = Ψ ∘ Ψ_{3}(u) ∈ U − Ψ(B(0, δ)) and therefore γ(Q) ∩ Ψ(B(0, δ)) = ∅.

Now, we claim that sup_{u∈Q} f(γ(u)) < c + ϵ. If u ∉ A, as before, we can assume that g(u) ∉ int(U_{1}) (and u ∈ Q); so we have two possibilities: g(u) ∈ U −U_{1} or g(u) ∉ U. In the first case, from Lemma 5.1 f(γ(u)) ≤ f(g(u)) < c + ϵ; in the second case, by definition of Φ, γ(u) = g(u), and therefore we have the claim. Finally, if u ∈ A, then f(γ(u)) = c − ||P_{−}(Ψ_{3}(u))]||^{2} < c + ϵ.

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Alberto Abbondandolo Scuola Normale Superiore, Piazza dei Cavalieri 7 56126 Pisa, Italy e-mail: abbo@sns.it Patricio Felmer Depto. de Ing. Matemática, Universidad de Chile Casilla 170 Correo 3, Santiago, Chile e-mail: pfelmer@dim.uchile.cl Juan Molina Depto. de Ing. Matemática, Universidad de Concepción Casilla 160 - C, Concepción, Chile e-mail: jmolina@ing-mat.udec.cl