# Minimax Results with Respect to Different Altitudes in the Situation of Linking

Authors: V. V. Motreanu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Consider a continuous function on a metric space. In the presence of linking between a compact pair and a closed set, depending on the different behaviors of the function on the linking sets, we establish minimax results guaranteeing existence of Palais-Smale sequences or providing gradient estimates. Our approach relies on deformation techniques.


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## Body

## 1. Introduction and Statement of Main Results

Minimax theorems play a central role in critical point theory: in this respect, we refer to celebrated minimax results as the mountain pass theorem (see Ambrosetti and Rabinowitz [1]) and the saddle point theorem (see Rabinowitz [2]). A practical way to define minimax values is by means of a linking condition in a topological space adapted to the problem. In the above-cited results, there is a specific linking based on the geometry of the involved problem. In this paper, we provide linking results for continuous functions on metric spaces, which extend the results in [3, 4] in this setting. Beyond this general framework, the main novelty that we emphasize with our approach is that we provide a systematic study related to the involved geometry, being concerned with situations which are not covered by the “classical geometry.” This approach is original in the case of Banach spaces and smooth functions too.Recall that when(X,∥·∥) is a Banach space and f:X→ℝ is continuously differentiable, an element u∈X is called a critical point of f if f′(u)=0. In this paper, more generally, we assume that X is a metric space endowed with the metric d and f is a continuous function. In this context, the following notion of critical point has been introduced by Degiovanni and Marzocchi (see [4]).Definition 1.
Letf:X→ℝ be a continuous function defined on a metric space X. An element u∈X is called a critical point of f if |df|(u)=0, where by |df|(u) we denote the supremum of the real numbers σ≥0 such that there exist δ>0 and a continuous map ℋ:Bδ(u)×[0,δ]→X satisfying
(1)d(ℋ(v,t),v)≤t,f(ℋ(v,t))≤f(v)-σt,∀(v,t)∈Bδ(u)×[0,δ].Remark 2.
(a) Note that|df|(u)∈[0,+∞] (take σ=0, any δ>0, and ℋ(v,t)≡v in (1)) and the map X→[0,+∞], u↦|df|(u) is lower semicontinuous. The quantity |df|(u) is usually called the weak slope of f at u.
(b) If(X,∥·∥) is a real Banach space, d is the distance induced by the norm ∥·∥, and f∈C1(X), then |df|(u) coincides with the norm of the differential f′(u). Thus, Definition 1 reduces to the usual notion of critical point in this context.Ifa∈ℝ, we denote by Ka the set of critical points of f at level a; that is,
(2)Ka:={u∈X:|df|(u)=0,f(u)=a}.The essential tool in our approach to critical point theory is the linking condition presented in the next definition. We say that(D,S) is a compact pair in X if S,D are compact subsets of X with S⊂D and D≠∅.Definition 3.
A compact pair(D,S) in X and a closed subset A of X are linking if the following property holds:
(3)S∩A=∅,γ(D)∩A≠∅,∀γ∈Γ,
where Γ:={γ∈C(D,X):γ|S=idS}.Given a continuous functionf:X→ℝ, a compact pair (D,S) in X, and a nonempty, closed subset A of X, we set
(4)a:=infAf,b:=supSf,c:=infγ∈ΓsupD(f∘γ).
Clearly, c≥b. Moreover, if (3) holds, then we have c≥a. Note that a,b,c∈ℝ∪{-∞}. If S≠∅, then b,c∈ℝ.The “classical” minimax principles are usually based on the assumption thatb<a (i.e., the “altitude” a-b is positive; actually, refined results involve the weaker condition b≤a), establishing in this situation the existence of a critical point for f. Our approach covers the situation where b>a (“negative altitude”). We will distinguish the cases: c=a and c>a. In the situation c>a, we will also distinguish the cases c=b>a and c>b.Now we formulate our main results. In all these results, unless otherwise stated,X is a metric space, f:X→ℝ is a continuous function, (D,S) is a compact pair in X and A is a closed subset of X satisfying (3).The first statement is a preliminary result providing existence and location of critical points.Proposition 4.
Assume thata>-∞ (see (4)) and that there exist γ∈Γ and a number ρ>0 such that
(5)f(γ(u))≤a,∀u∈Dwithγ(u)∈Bρ(A).
Then Ka∩A≠∅.Hereafter,Bρ(A)={u∈X:d(u,A)<ρ} (the open ρ-neighborhood of A). The proof of this proposition will be done in Section 2.In the following results, the metric spaceX is supposed to be complete.Here are our main results in the casesc=a and c>a.Theorem 5.
Assume thatc=a>-∞. Then
(6)inf{|df|(u):c≤f(u)<c+δ,u∈Bδ(A)}=0,∀δ>0.
In particular, there exists a sequence (un)⊂X such that
(7)|df|(un)⟶0,f(un)↓c,d(un,A)⟶0asn⟶∞.Theorem 6.
Assume thatc>a>-∞. Then(a)
for allδ>0 and ρ≥2d(S,A), one has
(8)inf{|df|(u):a≤f(u)<c+δ,u∈Bρ(A)}≤c-ad(S,A);(b)
for allδ>0 and ρ≥2d(S∩[f>a],A), one has
(9)inf{|df|(u):a≤f(u)<c+δ,u∈Bρ(A)}≤c-ad(S∩[f>a],A).Hereafter,[f>a]={u∈X:f(u)>a}. The notation d(S,A) stands for inf{d(u,v):u∈S,v∈A}.Remark 7.
The estimate in part (b) of Theorem6 is better than the one in part (a) when ρ∈[2d(S∩[f>a],A),+∞). Nevertheless, Theorem 6(a) provides an estimate for all ρ∈[2d(S,A),+∞).The next result distinguishes the situationsc>b and c=b in (4). Since Theorem 5 treats the case c=a, the next result is meaningful when c>a.Theorem 8.
(
a
)
Let (D,S) be a compact pair in X. Assume that c>b. Then
(10)inf{|df|(u):c≤f(u)<c+δ}=0,∀δ>0.
In particular, there exists a sequence (un)⊂X such that
(11)|df|(un)⟶0,f(un)↓casn⟶∞.
(
b
)
Let (D,S) be a compact pair in X and let A be a closed subset of X satisfying (3). Assume that c=b>a>-∞. Then
(12)inf{|df|(u):a≤f(u)<c+δ}≤b-ad(S∩[f>a],A),∀δ>0.As a consequence of Theorems5, 6(b), and 8, we have the next result, which studies the situations b≤a and b>a in (4).Corollary 9.
(
a
)
Assume that -∞<b≤a. Then (10) holds. If c=a, then (6) holds.
(
b
)
Assume that b>a>-∞. Then (12) holds. If c>b, then (10) holds.Theorems5 and 8(a) and Corollary 9(a) lead to the construction of a Palais-Smale sequence at level c (i.e., a sequence (un)⊂X such that f(un)→c and |df|(un)→0 as n→∞). Moreover, in the case of Theorem 5 (referring to the limiting casec=a), the Palais-Smale sequence is located near A (i.e., d(un,A)→0 as n→∞). Recall from [3] the following Palais-Smale condition.Definition 10.
We say that a continuous functionf:X→ℝ satisfies the Palais-Smale condition at level ℓ (condition (PS)ℓ for short), with ℓ∈ℝ, if every sequence (un)⊂X such that f(un)→ℓ and |df|(un)→0 as n→∞ has a convergent subsequence in X.If we assume condition(PS)c (see (4)), the previous results guarantee the existence of critical points.Corollary 11.
If conditions (3) and (PS)c hold, then one has the following.(a)
In the casec=a>-∞, then Kc∩A≠∅.(b)
In the casec>a, if one assumes that c>b (for instance, if b≤a), then Kc≠∅.Remark 12.
(a) Corollary11 actually holds under a weaker condition than (PS)c, namely, (PS)c+: every sequence (un)⊂X such that f(un)↓c and |df|(un)→0 as n→∞ has a convergent subsequence in X.
(b) Corollary11(a) holds under a weaker condition than (PS)c+, namely, (PS)c+,A: every sequence (un)⊂X such that f(un)↓c, d(un,A)→0, and |df|(un)→0 as n→∞ has a convergent subsequence in X. The conclusion in Corollary 11(a) is also more precise since it establishes the location of the critical point in the set A.Theorems6 and 8(b) and Corollary 9(b) go beyond the “classical geometry” b≤a in the situation of linking, allowing the case b>a and providing an estimate of the infimum of |df| at the points u with f(u)∈[a,c] in terms of b-a and of d(S,A) or of d(S∩[f>a],A).We may still obtain a Palais-Smale sequence from Theorem8(b) (the case of “nonclassical geometry”) under appropriate hypotheses for a sequence of sets (in the spirit of [5]) which are linking.Corollary 13.
Letf:X→ℝ be a continuous function on the metric space X, and let (An), (Dn), and (Sn) be sequences of subsets of X such that for all n≥1, An is closed, (Dn,Sn) is a compact pair in X, (Dn,Sn) and An satisfy (3) with Γ replaced by Γn:={γ∈C(Dn,X):γ|Sn=idSn}. Assume that
(13)an:=infAnf∈ℝ,bn:=supSnf∈ℝ,cn≔infγ∈ΓnsupDn(f∘γ)∈ℝ,∀n≥1,limn→∞max{bn-an,0}d(Sn∩[f>an],An)=0.
Then, denoting a-:=liminfn→∞an and c-:=limsupn→∞cn, there exists a sequence (un)⊂X satisfying
(14)a-≤liminfn→∞f(un)≤limsupn→∞f(un)≤c-,limn→∞|df|(un)=0.
In particular, if f satisfies condition (PS)ℓ for all ℓ∈[a-,c-], then there exists u∈X such that f(u)∈[a-,c-] and |df|(u)=0.Remark 14 (assume thatb≤a).
Considering particular situations of linking as in (3) in a Banach space (X,∥·∥), we obtain through Corollary 11 generalizations of classical minimax results in smooth critical point theory.(a)
IfD={te:t∈[0,1]}, S={0,e}, and A={u∈X:∥u∥=ρ} (with ρ>0 and e∈X such that ∥e∥>ρ), then we get the mountain pass theorem (see [1]).(b)
IfX=X1⊕X2, with dimX1<+∞, D={u∈X1:∥u∥≤1}, S={u∈X1:∥u∥=1}, and A=X2, then we get the saddle point theorem (see [2]).(c)
IfX=X1⊕X2, with dimX1<+∞, D={u∈X1:∥u∥≤r}  +  {te:t∈[0,r]},S=∂D, and A={u∈X2:∥u∥=ρ} (with r>ρ>0 and e∈X2 such that ∥e∥=1), we get the generalized mountain pass theorem (see [2]).
It is worth noting that generalizations of the minimax results cited in (a)–(c), relative to critical point theories for certain classes of nondifferentiable functionals, can already be found in the literature and have numerous applications to partial differential equations, differential inclusions, variational inequalities, hemivariational inequalities, and variational-hemivariational inequalities. In this respect, motivated by the study of existence of solutions of the so-called variational-hemivariational inequalities, introduced by Motreanu-Panagiotopoulos [6], we mention the critical point theory developed by these authors in [6] for functionals f=φ+ψ:X→ℝ∪{+∞}, with φ:X→ℝ locally Lipschitz and ψ:X→ℝ∪{+∞} lower semicontinuous, convex, and not identically +∞ (see also Chang [7] for the case when ψ=0, and see Szulkin [8] for the case when φ:X→ℝ is of class C1 and ψ:X→ℝ∪{+∞} is lower semicontinuous, convex, and not identically +∞).The rest of the paper is organized as follows. Section2 contains the proof of Proposition 4. Section 3 deals with the proofs of Theorems 5, 6, and 8 and Corollaries 9, 11, and 13. The method of proofs is based on suitable deformation results given in Sections 2, 3.


## 2. Proof of Proposition4

We start with stating the following deformation result.Theorem 15.
LetX be a metric space endowed with the metric d, and let f:X→ℝ be a continuous function. For all ρ>0, there exists a deformation η:X×[0,1]→X (i.e., η is continuous and satisfies η(u,0)=u for all u∈X) such that for all u∈X and t∈[0,1], one has(a)
t∈(0,1]⇒d(η(u,t),u)<ρt;(b)
|df|(u)=0⇒η(u,t)=u;(c)
|df|(u)>0, t∈(0,1]⇒f(η(u,t))<f(u).Proof.
First, assume that|df|≢+∞. We consider the continuous function σ:X→[0,+∞) given by
(15)σ(u):=12inf{d(u,v)+|df|(v):v∈X}.
We have σ(u)=0 for all u∈X with |df|(u)=0. Using that u↦|df|(u) is lower semicontinuous (see Remark 2(a)), we see that
(16)|df|(u)≠0⟹|df|(u)>σ(u)>0.
Applying [3, Theorem (2.8)], we find a deformation η^:X×[0,1]→X and a continuous map τ^:X→[0,+∞) with the properties (for all u∈X and t∈[0,1]): (i) τ^(u)=0⇔|df|(u)=0; (ii) d(η^(u,t),u)≤t; (iii) 0≤t≤τ^(u)⇒f(η^(u,t))≤f(u)-σ(u)t. Then the deformation η:X×[0,1]→X defined by
(17)η(u,t):=η^(u,12min{τ^(u),ρ}t),∀u∈X,∀t∈[0,1],
has properties (a)–(c) of the statement. If |df|≡+∞, we make the same reasoning taking an arbitrary positive constant in place of σ.Proof of Proposition4.
Letρ-:=min{d(S,A),ρ}>0, and let η:X×[0,1]→X be a deformation satisfying properties (a)–(c) of Theorem 15 (with ρ- in place of ρ). Note that S⊂X∖Bρ-(A). Let ϑ:X→[0,1] be a continuous function such that
(18)ϑ(u)=0⟺u∈X∖Bρ-(A).
Defining γ~:D→X by
(19)γ~(u):=η(γ(u),ϑ(γ(u))),∀u∈D,
with γ as in the statement of Proposition 4, we have γ~∈Γ. Fix u0∈D with γ~(u0)∈A (see (3)). Then, using property (a) in Theorem 15, we infer that
(20)d(γ(u0),A)≤d(γ(u0),γ~(u0))<ρ-≤ρ.
Hence, using properties (b), (c) of Theorem 15, relation (5), and the assumption that a∈ℝ, we obtain that
(21)a≤f(γ~(u0))=f(η(γ(u0),ϑ(γ(u0))))≤f(γ(u0))≤a
and |df|(γ(u0))=0. This combined with property (b) of Theorem 15 yields γ~(u0)=γ(u0)∈Ka∩A, which completes the proof.


## 3. Proofs of Theorems5, 6, and 8 and Corollaries 9, 11, and 13

The following deformation result was shown in [9].Theorem 16.
LetX be a metric space endowed with the metric d, let f:X→ℝ be a continuous function, let C be a subset of X, and let ℓ∈ℝ,σ>0, and ρ>0 be real numbers. Assume that the metric space f-1([ℓ,s]) (endowed with the induced metric) is complete for all s∈(ℓ,ℓ+σρ) and that
(22)|df|(u)>σ,∀u∈Bρ(C)∩[ℓ<f<ℓ+σρ].
Then there exist a continuous function τ:C∩[f<ℓ+σρ]→[0,ρ) and a continuous map η:(C∩[f<ℓ+σρ])  ×  [0,1]  →  [f<ℓ+σρ] such that for all u∈C∩[f<ℓ+σρ] and t∈[0,1], one has(a)
τ(u)≤(1/σ)max{f(u)-ℓ,0};(b)
d(η(u,t),u)≤τ(u)t;(c)
f(η(u,t))≤f(u)-στ(u)t;(d)
f(u)≥ℓ⇒f(η(u,1))=ℓ.Hereafter, we denote[f<s]={u∈X:f(u)<s} and [ℓ<f<s]={u∈X:ℓ<f(u)<s}.Proof of Theorem5.
For a fixedδ>0, it suffices to prove that
(23)inf{|df|(u):u∈[c<f<c+δ]∩Bδ(A)}>0⟹Ka∩A≠∅.
To this end, let σ be a number such that
(24)|df|(u)>σ,∀u∈[c<f<c+δ]∩Bδ(A).
Let ρ>0 such that
(25)σρ≤δ,4ρ≤min{d(S,A),δ}.
In particular, these choices combined with (24) guarantee that
(26)|df|(u)>σ,∀u∈Bρ(B-3ρ(A))∩[c<f<c+σρ].
Hereafter, we denote B-3ρ(A)={u∈X:d(u,A)≤3ρ}. Thus, applying Theorem 16 with the numbers ρ and σ, C:=B-3ρ(A), and ℓ:=c, we find a deformation η~:(C∩[f<c+σρ])×[0,1]→[f<c+σρ] such that
(27)d(η~(u,t),u)<ρ,f(η~(u,1))≤c=a,∀u∈C∩[f<c+σρ],∀t∈[0,1].
Using that3ρ≤d(S,A), we see that S⊂(X∖B3ρ(A))∩[f<c+σρ]. Let ϑ:[f<c+σρ]→[0,1] be a continuous function such that ϑ(u)=0 if u∈X∖B3ρ(A) and ϑ(u)=1 if u∈B-2ρ(A). Define the deformation η:[f<c+σρ]  ×  [0,1]  →  [f<c+σρ] by
(28)η(u,t):={η~(u,ϑ(u)t)ifu∈B3ρ(A),uotherwise.
To prove the continuity of η, we note that the restriction of η to the sets ([f<c+σρ]∖B3ρ(A))×[0,1] and ([f<c+σρ]∩B-3ρ(A))×[0,1], which are closed in [f<c+σρ], is continuous. Indeed, the restriction of η to ([f<c+σρ]∖B3ρ(A))×[0,1] coincides with the projection (u,t)↦u, whereas the restriction of η to ([f<c+σρ]×B-3ρ(A))×[0,1] coincides with the map (u,t)↦η~(u,ϑ(u)t) (since ϑ≡0 on ∂B3ρ(A):=B-3ρ(A)∖B3ρ(A)) which is continuous (by the continuity of ϑ).
Letγ0∈Γ with maxD(f∘γ0)<c+σρ. Then the map γ:D→X defined by
(29)γ(u):=η(γ0(u),1)
is well defined and γ∈Γ. Finally, let u∈D such that γ(u)∈Bρ(A). Since
(30)d(γ(u),γ0(u))=d(η(γ0(u),1),γ0(u))<ρ
(by the first part of (27)), we have that γ0(u)∈B2ρ(A), so ϑ(γ0(u))=1; thus
(31)f(γ(u))=f(η(γ0(u),1))=f(η~(γ0(u),1))≤a
(by the second part of (27)). Consequently, we can apply Proposition 4. Therefore, Ka∩A≠∅. This concludes the proof.Proof of Theorem6.
(a) By (3), we see that d(S,A)>0. For a fixed δ>0 and ρ-≥2d(S,A), it suffices to prove that
(32)inf{|df|(u):u∈[a<f<c+δ]∩Bρ-(A)}>c-ad(S,A)⟹Ka∩A≠∅.
To this end, let σ be a number such that
(33)|df|(u)>σ>c-ad(S,A),∀u∈[a<f<c+δ]∩Bρ-(A).
Let ρ,ε be positive numbers such that
(34)c-aσ<ρ<c-a+δσ,ρ+2ε<d(S,A).
Setting θ:=a+σρ-c∈(0,δ), since 2(ρ+ε)<ρ-, we have
(35)|df|(u)>σ,∀u∈Bρ(B-ρ+2ε(A))∩[a<f<c+θ].
This allows us to apply Theorem 16 with the numbers ρ and σ, C:=B-ρ+2ε(A), and ℓ:=a. Hence we find a deformation η~:(C∩[f<c+θ])  ×  [0,1]  →  [f<c+θ] such that
(36)d(η~(u,t),u)<ρ,f(η~(u,1))≤a,∀u∈C∩[f<c+θ],∀t∈[0,1].
The fact thatθ>0 in conjunction with the inequality ρ+2ε<d(S,A) yields S⊂(X∖Bρ+2ε(A))∩[f<c+θ]. Let ϑ:[f<c+θ]→[0,1] be a continuous function such that ϑ(u)=0 if u∈X∖Bρ+2ε(A) and ϑ(u)=1 if u∈B-ρ+ε(A). Define the deformation η:[f<c+θ]  ×  [0,1]  →[f<c+θ] by
(37)η(u,t):={η~(u,ϑ(u)t)ifu∈Bρ+2ε(A)uotherwise.
The continuity of η can be shown as in the proof of Theorem 5. Let γ0∈Γ with maxD(f∘γ0)<c+θ. Then the map γ:D→X given by
(38)γ(u):=η(γ0(u),1),∀u∈D,
is well defined, and we have γ∈Γ. Let u∈D such that γ(u)∈Bε(A). Since
(39)d(γ(u),γ0(u))=d(η(γ0(u),1),γ0(u))<ρ,
it follows that γ0(u)∈Bρ+ε(A), which yields
(40)f(γ(u))=f(η(γ0(u),1))=f(η~(γ0(u),1))≤a.
Consequently, Proposition 4 (with ρ replaced by ε) can be applied, and we conclude that Ka∩A≠∅.
(b) We see thatd(S∩[f>a],A)≥d(S,A)>0. For a fixed δ>0 and ρ-≥2d(S∩[f>a],A), it suffices to show that
(41)inf{|df|(u):u∈[a<f<c+δ]∩Bρ-(A)}>c-ad(S∩[f>a],A)⟹Ka∩A≠∅.
To this end, let σ be a number such that
(42)|df|(u)>σ>c-ad(S∩[f>a],A),∀u∈[a<f<c+δ]∩Bρ-(A).
Let ρ,ε be positive numbers such that
(43)c-aσ<ρ<c-a+δσ,ρ+2ε<d(S∩[f>a],A).
Setting θ:=a+σρ-c∈(0,δ), we have
(44)|df|(u)>σ,∀u∈Bρ(B-ρ+2ε(A))∩[a<f<c+θ].
Thus, we can apply Theorem 16 with the numbers ρ and σ, C:=B-ρ+2ε(A), and ℓ:=a. So we find a deformation η~:(C∩[f<c+θ])×[0,1]→[f<c+θ] such that
(45)η~(u,t)=u,∀(u,t)∈(C∩[f<c+θ])×[0,1]withf(u)≤a,(46)d(η~(u,t),u)<ρ,f(η~(u,1))≤a,∀u∈C∩[f<c+θ],∀t∈[0,1].
Letϑ:[f<c+θ]→[0,1] be a continuous function such that ϑ(u)=0 if u∈X∖Bρ+2ε(A) and ϑ(u)=1 if u∈B-ρ+ε(A). Define the deformation η:[f<c+θ]  ×  [0,1]  →  [f<c+θ] by
(47)η(u,t):={η~(u,ϑ(u)t)ifu∈Bρ+2ε(A),uotherwise.
The continuity of η can be shown as in the proof of Theorem 5.
Letγ0∈Γ with maxD(f∘γ0)<c+θ. Define γ:D→X by
(48)γ(u):={η(γ0(u),1)iff(γ0(u))>a,γ0(u)otherwise.
We show that γ∈Γ. To prove that γ is continuous, it suffices to note that the restrictions of γ to the closed sets {u∈D:f(γ0(u))≥a} and {u∈D:f(γ0(u))≤a} are continuous. This is immediate for the latter closed set. Concerning the former, this follows from the fact that the equality γ(u)=η(γ0(u),1) holds whenever f(γ0(u))=a (which is a consequence of the definition of η and relation (45)). Whence γ is continuous. Using that S∩[f>a]⊂X∖B-ρ+2ε(A), we infer that γ∈Γ.
Let an arbitrary pointu∈D with d(γ(u),A)<ε. If f(γ0(u))>a, then
(49)d(γ(u),γ0(u))=d(η(γ0(u),1),γ0(u))<ρ
(by the first part of (46)), so γ0(u)∈Bρ+ε(A), and thus f(γ(u))=f(η(γ0(u),1))=f(η~(γ0(u),1))≤a (by the second part of (46)). If f(γ0(u))≤a, then it is clear that f(γ(u))≤a. Consequently, in both cases, we have f(γ(u))≤a. This allows us to apply Proposition 4 (with ρ replaced by ε) and thus to obtain that Ka∩A≠∅, which completes the proof.Proof of Theorem8.
(a) The hypotheses allow us to apply Theorem5 with A:={u∈X:f(u)≥c}. Indeed, since c>b, we have that S∩A=∅. Moreover, since D is compact, for each γ∈Γ, there exists uγ∈D with maxD(f∘γ)=f(γ(uγ))≥c. It follows that condition (3) is satisfied and
(50)c=infγ∈Γf(γ(uγ))≥infAf≥c,
so infAf=c>-∞. The conclusion follows now from Theorem 5.
(b) This follows from Theorem6(b).Proof of Corollary9.
(a) Ifc>a, then we have c>b. Thus, we can apply Theorem 8(a). In the case c=a, we apply Theorem 5.
(b) Ifc>b, then we obtain from Theorem 8(a) that (10) holds (so that a fortiori (12) holds since b-a≥0). If c=b, then we apply Theorem 8(b).Proof of Corollary11.
Parts (a) and (b) follow from Theorems5 and 8(a), respectively, taking into account the lower semicontinuity of the map u↦|df|(u) (see Remark 2(a)).Proof of Corollary13.
Fix an arbitraryn≥1. If bn≤an, then applying Corollary 9(a), we have
(51)inf{|df|(u):cn≤f(u)<cn+1n}=0.
If bn>an, then using Corollary 9(b), we have
(52)inf{|df|(u):an≤f(u)<cn+1n}≤bn-and(Sn∩[f>an],An).
Let
(53)σn:=max{bn-an,0}d(Sn∩[f>an],An).
Then we have σn→0 as n→∞ and
(54)inf{|df|(u):an≤f(u)<cn+1n}≤σn,∀n≥1.
This completes the proof.



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*Source: 781746-2013-09-29.xml*