Mathematical Notes, vol. 77, no. 3, 2005, pp. 348–363. Translated from Matematicheskie Zametki, vol. 77, no. 3, 2005, pp. 378–394. c Original Russian Text Copyright 2005 by A. S. Leonov.
On the H-Property of Functionals in Sobolev Spaces A. S. Leonov Received June 27, 2002
Abstract—In this paper, we consider special classes of strongly convex functionals in Sobolev spaces. It is proved that functionals from such classes have the so-called H -property: weak convergence of sequences of arguments and convergence of such sequences with respect to a given functional imply strong convergence. H-property of functionals, strongly convex functional, weak and strong lower semicontinuity, Sobolev space, Banach space.
Key words:
1. INTRODUCTION Suppose that Z is a Banach space with elements z and D , D ⊂ Z , is a nonempty set. Suppose that a functional Ω(z) is defined on D . Definition 1.1. A functional Ω(z) is said to have the H -property on the set D if, for any sequence {zn } ⊂ D , its weak convergence as n → ∞ to some limit z0 ∈ D (denoted by zn z0 ) and convergence with respect to the functional Ω(z) (i.e., Ω(zn ) → Ω(z0 )), or Ω -convergence for short, yields strong convergence ( zn → z0 ). A typical example of a functional with H -property is the norm in the Hilbert space as well as norms in “spaces with H -property” (see, for example, [1]) in which, by definition, the convergence zn z0 and the convergence zn → z0 yield the strong convergence zn → z0 for each {zn } ⊂ Z . In particular, the Lebesgue spaces Lp , p > 1 , and the Sobolev spaces Wpl , p > 1 , l ≥ 1 (see [2]), and their norms have the H -property. Examples of functionals with the H -property in the space L1 were given in [3]. It follows from the results of [3] (Theorems 2.5 and 3.2) that integral functionals of the form 1
Ω(z) = 0
f [z(x)] dx
(1.1)
have the H -property on the set D+ = {z(x) ≥ 0, Ω(z) < ∞} of the space L1 [0, 1] if we have f (z) ∈ C[0, +∞) ∩ C 2 (0, +∞) , the derivative f (z) > 0 is monotone for z > 0 , and there exists a number ρ , 0 ≤ ρ < 1 , such that f (z)z ρ ≥ σ = const > 0 for all z > 0 . We can also give examples of functionals that do no have the H -property. Thus, in the space L2 [0, 1] , an integral functional of the form (1.1) with f (z) = |z|2 /(1 + |z|) does not have the H -property in the whole space. We can easily verify this by considering the sequence √ n sin(n), 0 ≤ x ≤ 1/n, zn (x) = 0, 1/n < x ≤ 1. This sequence is weakly convergent to zero in L2 [0, 1] and, as is readily seen, it converges to the functional (1.1): Ω(zn ) → Ω(0) . Nevertheless, it is not strongly convergent to zero, because 348
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zn − 02L2 = sin2 (n) . Note that Ω(z) ≤ z2L2 in this example. We can also give an example of a functional (1.1) satisfying the inequalities kz2L2 ≤ Ω(z) ≤ z2L2 k = const > 0 , but not having the H-property in L2 [0, 1] . In this connection, the following question arises: What functionals, in addition to the ones indicated above, have the H -property and how to describe the classes of these functionals in specific spaces (for example, in Sobolev spaces)? The present paper is concerned with the answer to this question. 2. STRONGLY CONVEX FUNCTIONALS Definition 2.1. A functional Ω(z) is said to be strongly convex on a convex set D , D ⊂ Z , if for its there exists a functional ∆(z1 , z2 ) ≥ 0 on D × D not identically zero such that z1 + z2 Ω(z1 ) + Ω(z2 ) ≥Ω + ∆(z1 , z2 ) 2 2
∀ z1 , z2 ∈ D.
(2.1)
Note that if condition (2.1) is satisfied with ∆(z1 , z2 ) = ξ(z1 − z2 ) , where ξ(t) is a continuous and monotone increasing (for t ≥ 0) function, with ξ(0) = 0 , then such a functional Ω is said to be (strictly) uniformly convex on D (see, for example, [4]). Definition 2.2. A functional ∆(z1 , z2 ) is said to have the S-property on the set D if, for any sequence {zn } ⊂ D , the convergence zn z0 ∈ D and the convergence ∆(zn , z0 ) → 0 as n → ∞ yield the strong convergence zn → z0 . Lemma 2.3. Suppose that a strongly convex functional Ω(z) on a convex set D is weakly lower semicontinuous on this set and the corresponding functional ∆(z1 , z2 ) from Definition 2.1 has the S -property. Then the functional Ω(z) has the H -property on D . Proof. Suppose that an arbitrary sequence {zn } ⊂ D weakly converges to an element z0 ∈ D and Ω -converges to this element (i.e., converges with respect to the functional Ω). Then, by the definition of the weak lower semicontinuity of the functional Ω(z) , we obtain the limiting relation: zn + z 0 ≥ Ω(z0 ). lim inf Ω n→∞ 2 Applying it and invoking the Ω -convergence ( Ω(zn ) → Ω(z0 )) in inequality (2.1) written for z1 = zn , z2 = z0 , we obtain
Ω(zn ) + Ω(z0 ) zn + z 0 ≥ lim Ω + ∆(zn , z0 ) Ω(z0 ) = lim n→∞ n→∞ 2 2 zn + z 0 + lim ∆(zn , z0 ) ≥ Ω(z0 ) + lim ∆(zn , z0 ). ≥ lim inf Ω n→∞ n→∞ n→∞ 2 This yields the convergence ∆(zn , z0 ) → 0 . This, together with the S -property of the functional ∆ , implies the strong convergence zn → z0 . This proves that the functional Ω has the H -property. The result of Lemma 2.3 is valid, for example, for weakly lower semicontinuous and strictly uniformly convex functionals, since the corresponding functional ∆(z1 , z2 ) = ξ(z1 −z2 ) obviously has the S -property. Some other examples are given below.
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3. STRONGLY CONVEX FUNCTIONALS WITH THE H -PROPERTY IN SOBOLEV SPACES In what follows, we shall assume that Z is the Sobolev space Wpl (T ) , p > 1 , l ≥ 1 , of functions z(x) defined in a closed bounded domain T ⊂ RN with sufficiently smooth boundary (see [2]); the norm on this space is given by
z = zWpl =
0≤|α|≤l
α
T
p
1/p
|D z(x)| dx
=
D
α
zpp
1/p .
(3.1)
0≤|α|≤l
Here Dα z(x) =
∂ |α| z , N · · · ∂xα N
1 ∂xα 1
|α| = α1 + · · · + αN ,
and
zp =
p
T
α1 , . . . , αN = 0, 1, . . . ,
1/p
|z(x)| dx
is the norm on the space Lp (T ) . The number of summands in the sum (3.1) will be denoted by ν . Suppose that linear operators of the form A0 z(x) = z(x),
Ak z(x) =
akα Dα z(x),
k = 1, . . . , m,
0≤|α|≤l
with continuous (in T ) coefficients akα = akα (x) are defined. Moreover, we assume that the following condition is satisfied: m k=0
Ak zpp ≥ c0 zpW l , p
c0 = const > 0,
z ∈ Wpl (T ).
(3.2)
First, consider the following functionals of special form defined on a given convex set D ⊂ Wpl (T ): Ω(z) = Ω[z(x)] = T
f x, |A0 z(x)|, |A1 z(x)|, . . . , |Am z(x)| dx,
z(x) ∈ D.
(3.3)
In what follows, we assume that the generating function f (x, y) = f (x, y0 , y1 , . . . , ym ) of this functional has the following properties: = {y0 ≥ 0, y1 ≥ (a) it is defined and continuous for x ∈ T , y = (y0 , y1 , . . . , ym ) ∈ Rm+1 + 0, . . . , ym ≥ 0} ; (b) it is nondecreasing in each variable yk , k = 0, . . . , m , given fixed other variables having admissible values; for each x ∈ T . (c) it is convex with respect to y in Rm+1 + Note that it follows from properties (a)–(c) that this function is bounded below. Therefore, without loss of generality, we can assume that f (x, y) ≥ 0 . Lemma 3.1. If the generating function f (x, y) has properties (a)–(c), then the functional (3.3) is weakly lower semicontinuous on a convex set D ⊂ Wpl (T ) .
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Proof. First, note that, by conditions (a)–(c), the function f (x, |y|) = f (x, |y0 |, |y1 |, . . . , |ym |) is continuous and convex with respect to y in Rm+1 for each x ∈ T . Therefore, on the set D , the following condition of the convexity of the functional (3.3) is satisfied: Ω(z1 ) + Ω(z2 ) 2 f (x, |z1 (x)|, |A1 z1 (x)|, . . . , |Am z1 (x)|) f (x, |z2 (x)|, |A1 z2 (x)|, . . . , |Am z2 (x)|) + dx = 2 2 T z1 (x) + z2 (x) , A1 z1 (x) + z2 (x) , . . . , Am z1 (x) + z2 (x) dx f x, ≥ 2 2 2 T z1 + z2 ∀ z 1 , z2 ∈ D (3.4) =Ω 2 and, moreover, Ω(z) ≥ 0 . We now prove the strong lower semicontinuity of this functional on the set D of the space Wpl (T ) . Choose any sequence {zn } ⊂ D so that zn → z0 ∈ D as n → ∞ . If, at the same time, lim inf Ω(zn ) = +∞, n→∞
then lim inf Ω(zn ) ≥ Ω(z0 ), n→∞
and the condition of strong lower semicontinuity is satisfied. But if the limit s = lim inf Ω(zn ) n→∞
is finite, then there exists a subsequence {zni } ⊂ {zn } for which lim Ω(zni ) = s
i→∞
and, moreover, zni → z0 , Dα zni → Dα z0 for all α , |α| = 1, . . . , l , almost everywhere in T . The last two convergences we can be ensured, since it follows from the condition zn − z0 Wpl → 0 that Dα zn → Dα z0 strongly in Lp (T ) for each α , |α| = 0, 1, . . . , l . Thus, the convergence
f x, |zni (x)|, |A1 zni (x)|, . . . , |Am zni (x)| → f x, |z0 (x)|, |A1 z0 (x)|, . . . , |Am z0 (x)| occurs almost everywhere in T . Combining this with the boundedness of the sequence {Ω(zni )} and using Fatou’s lemma, we obtain
f x, |z0 (x)|, |A1 z0 (x)|, . . . , |Am z0 (x)| dx T
f x, |zni (x)|, |A1 zni (x)|, . . . , |Am zni (x)| dx ≤ lim inf
Ω(z0 ) =
i→∞
T
= lim Ω(zni ) = lim inf Ω(zn ), i→∞
i→∞
i.e., and, in this case, the condition of strong lower semicontinuity of the functional (3.3) is satisfied. In conclusion, note that the weak lower semicontinuity of this functional follows from its strong lower semicontinuity and convexity on D [4]. MATHEMATICAL NOTES
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Theorem 3.2. Suppose that the generating function f (x, y) of the functional (3.3) has properties (a), (b) and, moreover, is strongly convex in the following sense: m u + v f (x, |u|) + f (x, |v|) |uk | − |vk | p ≥ f x, + κ 2 2
(3.5)
k=0
m+1
∀ x ∈ T , u, v ∈ R
,
κ = const > 0.
Then the corresponding functional (3.3) has the H -property on the set D . Proof. For the functional (3.3), from condition (3.5) with uk = Ak z1 (x) , vk = Ak z2 (x) for all z1 , z2 ∈ D , we can obtain, by analogy with (3.4), estimate (2.1) in which m |Ak z1 (x)| − |Ak z2 (x)| p dx. ∆(z1 , z2 ) = κ k=0
T
Therefore, the functional (3.3) turns out to be strongly convex. Let us show that such functional ∆(z1 , z2 ) has the S -property on D . Choose any sequence {zn } ⊂ D so that Wpl
z n z0 ∈ D
and
∆(zn , z0 ) → 0
as n → ∞ .
Then it follows from the form of the functional ∆ that Ak zn p → Ak z0 p
∀ k.
The weak convergence zn z0 in Wpl implies the weak convergence Lp
D α zn D α z0
∀ α,
0 ≤ |α| ≤ l,
Lp
and, further, Ak zn Ak z0 . It then follows that the H -property of the space Lp yields the Lp
relation Ak zn → Ak z0 . Combining it with the estimate m k=0
Ak zn − Ak z0 pp ≥ c0 zn − z0 pW l
p
following from condition (3.2) leads to the strong convergence zn → z0 in Wpl . Thus, we have found that the convergence zn z0 , ∆(zn , z0 ) → 0 implies the strong convergence zn → z0 . This means that the S -property holds for ∆(z1 , z2 ) . Since, by the assumptions of the theorem, the generating function f (x, y) has properties (a) and (b) and property (c) following from (3.5), applying Lemma 3.1, we see that the strongly convex functional (3.3) will be weakly lower semicontinuous. As a result, by Lemma 2.3, it will have the H -property on D . p m Remark 3.3. If, in condition (3.5) (in Theorem 3.2), the term k=0 |uk | − |vk | is replaced by m p k=0 |uk − vk | , then the corresponding functional (3.3) will also be strongly convex on D and, more precisely, strictly uniformly convex. For it, the result of Theorem 3.2 is also valid. Let us illustrate the application of Theorem 3.2 by a simple example. It is well known that the space Wpl (T ) has the H -property for p > 1 , l ≥ 1 , i.e., in it, the weak convergence zn z0 and the convergence of the norms zn → z0 always implies the strong convergence zn → z0 . MATHEMATICAL NOTES
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For p ≥ 2 , this assertion also follows from Theorem 3.2. Indeed, suppose that Ω(z) = zpW l and p p ≥ 2 . Taking into account the form of the generating function f (x, y) = f (y) = |y0 |p + |y1 |p + · · · + |yν |p of this functional (see (3.1)) and using the numerical inequality s1 + s2 p |s1 |p + |s2 |p + κ|s1 − s2 |p ≥ 2 2
∀ s1 , s2 ∈ R,
κ=
min{1, 23−p } , 8
p ≥ 2, (3.6)
we obtain ν ν ν u k + vk p f (u) + f (v) |uk |p + |vk |p = ≥ |uk − vk |p +κ 2 2 2 k=0 k=0 k=0 ν u+v |uk | − |vk | p ≥f ∀ u = (u0 , . . . , uν ), v = (v0 , . . . , vν ). +κ 2 k=0
This yields assumption (3.5) of Theorem 3.2. This theorem guarantees that the strongly (strictly uniformly) convex functional zpW l for p ≥ 2 has the H -property on the whole space Wpl (T ) . p
However, condition (3.5) and, in particular, the property of strict uniform convexity of the functionals is often not satisfied even in the simplest cases. For example, the same functional zpW l for 1 < p < 2 is no longer strictly uniformly convex on the whole space Wpl (T ) . The proof p
is similar to that of [4, p. 217] and is a consequence of the fact that, for 1 < p < 2 , inequalities of type (3.6) do not hold on the whole set R . Hence Theorem 3.2 and Remark 3.3 do not give an answer to the question about the H -property of the norm on Wpl (T ) in this case. Nevertheless, the space Wpl (T ) , and hence also the functional zpW l , has the H -property in the case 1 < p < 2 , p l ≥ 1. In the remaining part of this section, we shall study classes of functionals of general form in Wpl (T ) . For a more detailed study of the case 1 < p < 2 , we introduce the quantity ∆lp (z1 , z2 ) =
0≤|α|≤l
=
T
2 |Dα z1 (x)|2−p + |Dα z2 (x)|2−p |Dα z2 (x)|p−1 − |Dα z1 (x)|p−1 dx
∆p (Dα z1 (x), Dα z2 (x)).
(3.7)
0≤|α|≤l
Let us study it. Lemma 3.4. The functional ∆lp (z1 (x), z2 (x)) is defined for all functions z1 (x) , z2 (x) ∈ Wpl (T ) for 1 < p < 2 . Proof. The functional (3.7) consists of similar summands expressed via the quantity
2 |z1 (x)|2−p + |z2 (x)|2−p |z2 (x)|p−1 − |z1 (x)|p−1 dx, ∆p (z1 , z2 ) = T
in which the elements z1 (x), z2 (x) and Dα z1 (x), Dα z2 (x) of the space Lp (T ) appear. Therefore, it suffices to prove that ∆p (z1 , z2 ) is finite in Lp (T ) . Searching for the minimum of the function p |t|p + 1 t + 1 − − κ(|t|2−p + 1)(|t|p−1 − 1)2 , ζ(t) = 2 2 MATHEMATICAL NOTES
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we find that ζ(t) ≥ ζmin = ζ(1) = 0 . Noting t = s1 /s2 , s1 , s2 ∈ R , we obtain the numerical inequality s1 + s2 p
|s1 |p + |s2 |p + κ |s1 |2−p + |s2 |2−p |s2 |p−1 − |s1 |p−1 2 . ≥ (3.8) 2 2 Substituting the functions z1 (x), z2 (x) ∈ Lp (T ) , 1 < p < 2 , for s1 , s2 in this inequality, integrating over x , and using the properties of the Lebesgue integral, we obtain
z1 + z2 p
2 z1 pp + z2 pp
≥ +κ |z1 (x)|2−p + |z2 (x)|2−p |z2 (x)|p−1 − |z1 (x)|p−1 dx
2 2 T p
p
z1 + z2
= (3.9)
2 + κ∆p (z1 , z2 ). p Hence it follows that that ∆p (z1 , z2 ) is finite. Lemma 3.5. Let 1 < p < 2 . Suppose that, for a sequence {zn } ⊂ Lp (T ) and for an element z0 ∈ Lp (T ) , the limiting relation ∆p (zn , z0 ) → 0 , n → ∞ , holds. Then the following convergences hold : |zn (x)|p − |z0 (x)|p dx → 0, zn p → z0 p , n → ∞. T
Proof. We use the well-known numerical inequality p
|s2 | − |s1 |p ≤ p |s2 | + |s1 | |s2 |p−1 − |s1 |p−1 1 < p < 2, ∀ s1 , s2 ∈ R, p−1 to obtain the estimate
p p zn (x)pp − z0 (x)pp = ≤ |zn (x)|p − |z0 (x)|p dx |z dx (x)| − |z (x)| n 0 T T
p |zn (x)| + |z0 (x)| |zn (x)|p−1 − |z0 (x)|p−1 dx. ≤ p−1 T Using the Cauchy–Bunyakovskii inequality on its right-hand side, we obtain zn (x)pp − z0 (x)pp ≤ |zn (x)|p − |z0 (x)|p dx p ≤ p−1
T
2(1−p/2)
T
|zn (x)|
+ T
≤
|zn (x)|
2(1−p/2)
|z0 (x)|
p−1
p−1 2
− |z0 (x)|
1/2 dx
2 |zn (x)|p−1 − |z0 (x)|p−1 dx
|zn (x)| T 1/2
2 p |zn (x)|2−p + |z0 (x)|2−p |zn (x)|p−1 − |z0 (x)|p−1 dx p−1 T 1/2 1/2 p p × |zn (x)| dx + |z0 (x)| dx T
2(p/2)
T 1/2
1/2 dx
2(p/2)
|z0 (x)|
1/2 dx
T
1/2 p = + z0 (x)p/2 (3.10) ∆p (zn , z0 ). zn (x)p/2 p p p−1 Combining this with the assumptions of the lemma, we can write zn (x)p/2 ≤ p ∆1/2 − z0 (x)p/2 (zn , z0 ) → 0, (3.11) p p p−1 p i.e., we have zn p → z0 p as n → ∞ . Then from (3.10) and (3.11) we obtain the other convergence as well: |zn (x)|p − |z0 (x)|p dx → 0. T
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Corollary 3.6. If, for a sequence {zn } ⊂ Wpl (T ) , 1 < p < 2 , and for an element z0 ∈ Wpl (T ) , the limiting relation ∆lp (zn , z0 ) → 0 , n → ∞ , holds, then the following inequality is valid : √ zn p/2l − z0 p/2l ≤ p ν ∆1/2 (zn , z0 ), Wp Wp p − 1 lp
(3.12)
which implies zn Wpl → z0 Wpl . Moreover, T
|zn (x)|p − |z0 (x)|p dx → 0,
n → ∞.
Indeed, it is readily verified that the following inequality holds: p/2 p/2 p p α α zn l − z0 l = D zn p − D z0 p Wp Wp 0≤|α|≤l 0≤|α|≤l Dα zn p/2 . − Dα z0 p/2 ≤ p p 0≤|α|≤l
Proceeding with the task of estimating its right-hand side and using analogs of estimates (3.11) obtained above and the Cauchy–Bunyakovskii inequality, we obtain zn p/2l − z0 p/2l ≤ Wp
Wp
p p−1
α α ∆1/2 p (D zn , D z0 )
0≤|α|≤l
1/2 √ √ p ν 1/2 p ν α α ∆p (D zn , D z0 ) = ∆ (zn , z0 ) → 0. ≤ p−1 p − 1 lp 0≤|α|≤l
This proves the required convergence of the norms as n → ∞ . The second convergence to be proved can be obtained from Lemma 3.5. Definition 3.7. A functional of general form Ω is said to be quasiuniformly convex on a convex set D ⊂ Wpl (T ) , 1 < p < 2 , if there exists a continuous and monotone increasing function ξ(t) , t ≥ 0 , ξ(0) = 0 , such that z1 + z2 Ω(z1 ) + Ω(z2 ) ≥Ω + ξ[∆lp (z1 , z2 )] 2 2
∀ z1 , z2 ∈ D.
(3.13)
As an example of a quasiuniformly convex functional on the whole space Wpl (T ) , 1 < p < 2 , we can consider Ω(z) = zpW l . This can be established by the same scheme as the strict uniform p
convexity of the norm for p ≥ 2 . Indeed, in view of inequality (3.8), the generating function f (x, y) = f (y) = |y0 |p + |y1 |p + · · · + |yν |p of this functional has the property ν
2 f (u) + f (v) u+v |uk |2−p + |vk |2−p |uk |p−1 − |vk |p−1 ≥f +κ 2 2 k=0
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∀ u, v ∈ Rν+1 .
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Combining this with the Definition (3.1) of the norm, we obtain the condition of quasiuniform convexity with ξ(t) = κt: z1 pW l + z2 pW l |Dα z1 (x)|p + |Dα z2 (x)|p p p = dx 2 2 0≤|α|≤l T z1 + z2 p D α dx ≥ 2 T 0≤|α|≤l
2 |Dα z1 |2−p + |Dα z2 |2−p |Dα z1 |p−1 − |Dα z2 |p−1 dx +κ 0≤|α|≤l
T
z 1 + z 2 p
+ κ∆lp (z1 , z2 ).
= 2 W l p
Theorem 3.8. If a functional of general form Ω is quasiuniformly convex and weakly lower semicontinuous on a convex set D , then it has the H -property on D . Proof. The quasiuniform convexity of the functional Ω is a special case of strong convexity (see Definition 2.1), with ∆(z1 , z2 ) = ξ[∆lp (z1 , z2 )] . Suppose that, for an arbitrary sequence {zn } ⊂ D , the following relations are satisfied: zn z0 ∈ D,
∆(zn , z0 ) → 0,
n → ∞.
Then, from the properties of the function ξ(t) , we obtain ∆lp (zn , z0 ) = ξ −1 [∆(zn , z0 )] → 0,
n → ∞.
Therefore, by (3.12), we have zn Wpl → z0 Wpl . Using the weak convergence of the sequence, the convergence of the norms proved above, and the H -property of the space Wpl (T ) , we obtain the strong convergence zn → z0 . Thus, the functional ∆(z1 , z2 ) = ξ[∆lp (z1 , z2 )] has the S -property (using the terminology of Sec. 2). But, in that case, it follows from Lemma 2.3 that the functional Ω has the H -property on the set D . Next, we introduce more general classes of strongly convex functionals related to the fixed operators Ak z(x) . Definition 3.9. A functional Ω(z) is said to be uniformly convex in the extended sense on a convex set D , D ⊂ Wpl (T ) , if there exists a continuous and increasing the function ξ(t) , t ≥ 0 , ξ(0) = 0 , and a continuous and increasing function ϕ(s) , s ≥ 0 , such that, for all z1 , z2 ∈ D , the following inequality is valid: Ω(z1 ) + Ω(z2 ) z 1 + z2 ≥Ω + ξ(Φ(z1 , z2 )), (3.14) 2 2 where
m ϕ(|Ak z1 |) − ϕ(|Ak z2 |) p dx. Φ(z1 , z2 ) = k=0
T
A functional Ω is called quasiuniformly convex in the extended sense on a convex set D ⊂ Wpl (T ) , 1 < p < 2 , if, instead of (3.14), the following relation is satisfied: z1 + z2 Ω(z1 ) + Ω(z2 ) ≥Ω + ξ(Ψ(z1 , z2 )) ∀ z1 , z2 ∈ D, (3.15) 2 2 in which Ψ(z1 , z2 ) =
m
∆p [ϕ(|Ak z1 |), ϕ(|Ak z2 |)].
k=0
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Remark 3.10. Suppose that the set D contains a function z(x) for which ϕ(|Ak z(x)|) ∈ Lp (T ),
0 ≤ k ≤ m.
Then conditions (3.14) or (3.15) induce the inclusion ϕ(|Ak z(x)|) ∈ Lp (T ) for any z(x) ∈ D . Indeed, it follows from (3.14) that m p z) z + z −1 Ω(z) + Ω( −Ω < +∞ Φ(z , z) = ϕ(|Ak z|) − ϕ(|Ak z|) dx ≤ ξ 2 2 T k=0
for all z(x) ∈ D ⊂ Wpl (T ) , i.e., the function ϕ(|Ak z(x)|) − ϕ(|Ak z(x)|) , and hence ϕ(|Ak z(x)|) , belongs to Lp (T ) , 0 ≤ k ≤ m . But if condition (3.15) is satisfied, then, replacing zn (x), z0 (x) by ϕ(|Ak z(x)|), ϕ(|Ak z(x)|) in inequality (3.11) and carrying out the estimate as in the proof of Corollary 3.6, we obtain 1/2 √ m m
p/2
p/2 p m
∆p [ϕ(|Ak z1 |), ϕ(|Ak z2 |)] ϕ(|Ak z|) p − ϕ(|Ak z|) p ≤ p−1 k=0 k=0 1/2 √ √ z) z + z p m 1/2 p m −1 Ω(z) + Ω( Ψ (z , z) ≤ ξ −Ω = < +∞, p−1 p−1 2 2 which again proves the inclusion ϕ(|Ak z(x)|) ∈ Lp (T ) , 0 ≤ k ≤ m . Lemma 3.11. Suppose that a continuous and monotone increasing function ϕ(s) , s ≥ 0 , satisfies the inequalities ϕ(s) ≥ 0, ϕ(s) ≥ a + bs ∀ s ≥ 0, where a, b , b > 0 , are constants. Further, suppose that the sequence {zn (x)} ⊂ D and the element z0 (x) ∈ D satisfy the conditions ϕ(|zn (x)|) ∈ Lp (T ) and ϕ(|z0 (x)|) ∈ Lp (T ) . Then (1) if the convergence
ϕ(|zn (x)|) − ϕ(|z0 (x)|) → 0 as n → ∞ p takes place, then zn p → z0 p ; (2) if, for 1 < p < 2 , the limiting relation ∆p [ϕ(|zn (x)|), ϕ(|z0 (x)|)] → 0 is valid, then zn p → z0 p as well. Proof. In the first case, we have
ϕ(|zn (x)|) − ϕ(|z0 (x)|) → 0, p i.e.,
Lp
vn (x) = |vn (x)| = ϕ(|zn (x)|) → ϕ(|z0 (x)|) = |v0 (x)| = v0 (x) ∈ Lp (T ). Let us study the limiting relation between the functions |zn (x)| = ϕ−1 (|vn (x)|),
n → ∞,
and
|z0 (x)| = ϕ−1 (|v0 (x)|)
In view of the conditions on the function v = ϕ(s) , we obtain 0 ≤ s = ϕ−1 (v) = ϕ−1 (|v|) ≤ a0 + b0 v = a0 + b0 |v|, MATHEMATICAL NOTES
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where a0 = −a/b , b0 = 1/b . Thus, |ϕ−1 (|v|)| ≤ a0 + b0 |v| . By the results from [5, p. 204; 6, p. 312], this condition ensures that the operator ϕ−1 [ · ] , acting on the elements v(x) of the space Lp (T ) by the rule |z(x)| = ϕ−1 [|v(x)|] , is continuous from Lp to Lp . Therefore, Lp
|zn (x)| = ϕ−1 [|vn (x)|] → ϕ−1 [|v0 (x)|] = |z0 (x)| ; hence we obtain the convergence of the norms zn p → z0 p . In the second case, applying Lemma 3.5 with ϕ(|zn (x)|), ϕ(|z0 (x)|) , substituted for zn = zn (x) , z0 = z0 (x) , respectively, and taking the condition ϕ(s) ≥ 0 into account, we can write p ϕ (|zn (x)|) − ϕp (|z0 (x)|) dx → 0 as n → ∞. T
This implies
L
wn (x) = ϕp (|zn |) →1 ϕp (|z0 |) = w0 (x) ∈ L1 (T ). Further, we use the fact that the involution operator v(x) = |w(x)|1/p is continuous from L1 to Lp , as follows from [5, p. 204; 6, p. 312]. Therefore, 1/p Lp p 1/p → ϕ (|z0 (x)|) = ϕ(|z0 (x)|) = v0 (x). vn (x) = ϕ(|zn (x)|) = ϕp (|zn (x)|) The subsequent arguments using the conditions ϕ(s) ≥ 0 , ϕ(s) ≥ a + bs , are carried out just as in the proof of the first part of this lemma and again yield the convergence zn p → z0 p . Theorem 3.12. Suppose that for the function ϕ(s) appearing in Definition 3.9, the following conditions are satisfied : ϕ(s) ≥ 0,
ϕ(s) ≥ a + bs
∀ s ≥ 0,
a, b = const,
b > 0.
Wpl (T ) ,
contains a function z(x) such that Further, suppose that the convex set D , D ⊂ ϕ(|Ak z|) ∈ Lp (T ) for all k = 0, . . . , m (for example, z(x) = C0 = const ∈ D ) and the functional Ω(z) is weakly lower semicontinuous on D . Then (1) if the functional Ω(z) is uniformly convex in the extended sense on the set D , then it has the H -property; (2) but if the functional Ω(z) quasiuniformly convex in the extended sense on the convex set D ⊂ Wpl (T ) , 1 < p < 2 , then it also has the H -property. Proof. Generalized uniform convexity is a version of strong convexity for the functional Ω(z) with ∆(z1 , z2 ) = ξ[Φ(z1 , z2 )] . Take an arbitrary sequence {zn } ⊂ D with properties zn z0 ∈ D and ξ[Φ(zn , z0 )] → 0 as n → ∞ . Then m ϕ(|Ak zn |) − ϕ(|Ak z0 |) p dx → 0, k=0
T
and hence, by Lemma 3.11 (in view of Remark 3.10), we have Ak zn p → Ak z0 p
∀k
as
n → ∞.
Further, repeating the corresponding part of the proof of Theorem 3.2 word for word, we see that this convergence of the norms implies zn → z0 in Wpl (T ) , which, in turn, shows that the S -property for the functional ∆(z1 , z2 ) = ξ[Φ(z1 , z2 )] holds. By Lemma 2.3, the functional Ω(z) which is uniformly convex in the extended sense and weakly lower semicontinuous on D will then possess the H -property. Now, suppose that Ω(z) is a functional quasiuniformly convex in the extended sense on the set D ⊂ Wpl (T ) , 1 < p < 2 , Then it is strongly convex with ∆(z1 , z2 ) = ξ[Ψ(z1 , z2 )] . Again choosing an arbitrary sequence for which zn z0 ∈ D and ξ[Ψ(zn , z0 )] → 0 , we obtain Ψ(zn , z0 ) → 0 , i.e., ∆p [ϕ(|Ak zn |), ϕ(|Ak z0 |)] → 0 . Then, by Lemma 3.11, we have the convergence Ak zn p → Ak z0 p for all k . The subsequent proof is carried out as that for the case of generalized uniform convexity.
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4. SUFFICIENT CONDITIONS OF STRONG CONVEXITY OF FUNCTIONALS IN SOBOLEV SPACES Let us again turn to functionals of the form (3.3). For them to be uniformly convex in the extended sense or quasiuniformly convex in the extended sense on the convex set D ⊂ Wpl (T ) , it suffices, for example, to require that the generating function f (x, y) = f (x, y0 , y1 , . . . , ym ) satisfy conditions (a) and (b) as well as one of the following inequalities: m u + v f (x, |u|) + f (x, |v|) ϕ(|uk |) − ϕ(|vk |) p ≥ f x, +κ 2 2
(4.1)
∀ u, v ∈ Rm+1 , κ = const > 0, p > 1, m u + v
f (x, |u|) + f (x, |v|) 2−p 2−p +κ ≥ f x, (|u |) + ϕ (|v |) ϕ k k 2 2 k=0 p−1
2 × ϕ (|uk |) − ϕp−1 (|vk |) , 1 < p < 2,
(4.2)
k=0
∀x ∈ T,
where the function ϕ(s) satisfies the constraints from Definition 3.9 and the assumptions of Lemma 3.11 (Theorem 3.12). Then, on the one hand, the function f (x, y) satisfies conditions (a)– (c), and hence, by Lemma 3.1, the corresponding functional (3.3) is weakly lower semicontinuous. On the other hand, the functional (3.3) satisfies inequality (3.14) or (3.15) with ξ(t) = κt . This is readily verified just as in the proof of Theorem 3.2 in which conditions (3.5) are replaced by (4.1) or (4.2). Hence, by Theorem 3.12, such a functional (3.3) will have the H -property. For ϕ(s) = s , inequalities (4.1) and (4.2) likewise ensure the strict uniform and the quasiuniform convexity of the functional (3.3), respectively, and, by Theorems 3.2 and 3.8 also its H -property. However, we are interested in obtaining more detailed sufficient conditions on the generating function which ensure the H -property of the functionals, not just inequalities (4.1) and (4.2). Such conditions can be found for functionals (3.3) of special form Ω(z) =
T
f x, ϕp (|z(x)|), ϕp (|A1 z(x)|), . . . , ϕp (|Am z(x)|) dx,
(4.3)
defined on a convex set D ⊂ Wpl (T ) . Theorem 4.1. Suppose that and is (1) the function f (x, y) ≥ 0 is defined for x ∈ T , y = (y0 , y1 , . . . , ym ) ∈ Rm+1 + continuous there; (2) it is differentiable with respect to y for each x ∈ T and, moreover, the inequalities ; fy i (x, y) ≥ γ = const > 0 , i = 0, 1, . . . , m , hold for all (x, y) ∈ T × Rm+1 + m+1 for each x ∈ T ; (3) it is convex with respect to y in R+ (4) the function ϕ(s) is continuous, nonnegative, monotone increasing, and convex for s ≥ 0 . Then the functional (4.3) is quasiuniformly convex in the extended sense for 1 < p < 2 and uniformly convex in the extended sense for p ≥ 2 on D . Proof. Suppose that, for example, p ≥ 2 . Then, using the convexity of the function f (x, y) in y , the monotonicity of the function f (x, y) = f (x, y0 , y1 , . . . , ym ) in the arguments yk , k = 0, 1, . . . , m , (following from condition (2)), applying an inequality of type (3.6) with the numbers s1 , s2 replaced by ϕp (|uk |) and ϕp (|vk |) , k = 0, 1, . . . , m , denoting ϕp (|u|) = (ϕp (|u0 |), . . . , ϕp (|um |)), MATHEMATICAL NOTES
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and using the finite increment formula, we obtain ϕp (|u|) + ϕp (|v|) f (x, ϕp (|u|)) + f (x, ϕp (|v|)) ≥ f x, 2 2 p p ϕ(|u|) + ϕ(|v|) ≥ f x, + κ ϕ(|u|) − ϕ(|v|) 2 ϕ(|u|) + ϕ(|v|) p = f x, 2 m ϕ(|u|) + ϕ(|v|) p p p · κ ϕ(|uk |) − ϕ(|vk |) fyk x, + κθ ϕ(|u|) − ϕ(|v|) + 2 k=0
m ϕ(|u|) + ϕ(|v|) p ϕ(|uk |) − ϕ(|vk |) p + γκ ≥ f x, 2
k=0
∀ u = (u0 , u1 , . . . , um ), v = (v0 , v1 , . . . , vm ),
0 < θ < 1.
We continue our calculation by taking into account the following facts: the function f (x, y) is increasing in y and the function ϕ(s) ≥ 0 is convex and increasing for s ≥ 0: ϕ(|u|) + ϕ(|v|) p p |u| + |v| p u + v ≥ f x, ϕ ≥ f x, ϕ . f x, 2 2 2
As a result, for each x ∈ T and any u, v ∈ Rm+1 , we obtain m u + v f (x, ϕp (|u|)) + f (x, ϕp (|v|)) ϕ(|uk |) − ϕ(|vk |) p . ≥ f x, ϕp + γκ 2 2 k=0
Substituting Ak z1 (x), Ak z2 (x) , where z1 (x), z2 (x) are arbitrary functions from the convex set D ⊂ Wpl (T ) , for uk , vk in this numerical inequality and integrating the resulting relation term by term, we obtain the following version of formula (3.14): m 2) 1 ) + Ω(z z1 + z 2 Ω(z ϕ(|Ak z1 (x)|) − ϕ(|Ak z2 (x)|) p dx ≥Ω + γκ 2 2 T
(4.4)
k=0
for all z1 , z2 ∈ D . This proves the uniform convexity (in the extended sense) of the functional (4.3). In the case 1 < p < 2 , quasiuniform convexity (in the extended sense) is proved using the same scheme with some modifications. Thus, instead of inequality (3.6), we now use inequality (3.8) with the numbers s1 , s2 replaced by ϕ(|uk |), ϕ(|vk |) , respectively, and with the subsequent replacement in the calculations of the term |ϕ(|uk |) − ϕ(|vk |)|p by (ϕ2−p (|uk |) + ϕ2−p (|vk |))(ϕp−1 (|uk |) − ϕp−1 (|vk |))2 . In particular, this leads to a modification of inequality (4.4). As a result, we obtain the following version of relation (3.15): m 2) 1 ) + Ω(z z1 + z2 Ω(z ≥Ω + γκ ∆p [ϕ(|Ak z1 |), ϕ(|Ak z2 |)] 2 2
∀ z1 , z2 ∈ D,
k=0
whereby the theorem is proved. MATHEMATICAL NOTES
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Corollary 4.2. If the set D satisfies the assumptions of Theorem 3.12 (for example, contains a constant function) and the functions f and ϕ satisfy the assumptions of Theorem 4.1 and, moreover, just as in Theorem 3.12, the condition ϕ(s) ≥ a + bs ( b > 0 ) holds for s ≥ 0 , then the functional (4.3) is weakly lower semicontinuous on D and has the H -property there. Indeed, the generating function f0 (x, y) = f (x, ϕp (y)) of the functional (4.3) has properties (a)– (c) from Sec. 3. Its continuity and monotonicity follows from conditions (1), (2), and (4) of Theorem 4.1 and its convexity is a consequence of inequality (4.4) for p ≥ 2 or of its analog for 1 < p < 2 . Therefore, for a proof, it remains to apply Lemma 3.1, Theorem 4.1, and Theorem 3.12. Consider some examples. As is readily verified, in view of Theorem 4.1, the following functionals with ϕ(s) constrained by the assumptions of Theorem 3.12 turn out to be uniformly convex in the extended sense for p ≥ 2 and quasiuniformly convex in the extended sense for 1 < p < 2 on D: m q(x) exp[ϕp (|Ak z(x)|)] dx, Ω1 (z) = Ω2 (z) =
k=0 T m k=0
T
q(x)[ϕp (|Ak z(x)|) + 1] ln[ϕp (|Ak z(x)|) + 1] dx.
Here q(x) ≥ q0 = const > 0 is a continuous function defined in T . By Corollary 4.2, since the functionals Ω1,2 (z) are weakly lower semicontinuous, they have the H -property on D . Of particular interest is the functional m p |Ak z(x)| + r p dx, (4.5) q(x)[|Ak z(x)| + r] ln Ω3 (z) = r T k=0
defined on a convex set D ⊂ Wpl (T ) . Here r > 0 is a given constant. For Z = L1 (T ) , the functional z(x) ΩE (z) = z(x) ln dx, r T whose form is close to that of the functional (4.5), defined on Dr = {z(x) ∈ L1 (T ) : z(x) ≥ r, ΩE (z) < ∞} and used in the so-called maximal entropy method is well studied (see, for example, [7, 8]). In particular, its H -property on Dr was actually established in [3]. The functional (4.5) has not been studied earlier from this point of view. The generating function of the functional (4.5) f (x, y) = q(x)
m
yk
k=0
satisfies the assumptions of Theorem 4.1 on the set T × Rm+1 . The function s+r , s ≥ 0, ϕ(s) = (s + r) ln r also satisfies the assumptions of Theorem 4.1. Moreover, ϕ(0) = 0 and ϕ(s) ≥ s . Therefore, by Theorem 4.1, the functional Ω3 (z) is uniformly convex in the extended sense for p ≥ 2 or quasiuniformly convex in the extended sense for 1 < p < 2 on the set D . By Corollary 4.2, it will be weakly lower semicontinuous on D and will have the H -property there. Now, let us consider functionals of the form m Ω(z) = q(x)|Ak z(x)|p ψ(|Ak z(x)|) dx, z(x) ∈ D ⊂ Wpl (T ), (4.6) k=0
T
where q(x), ψ(z) are given functions. Such functionals are often encountered in applications. MATHEMATICAL NOTES
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Theorem 4.3. Suppose that q(x) ≥ q0 = const > 0 is a continuous function in T and ψ(s) , s ≥ 0 , is a nonnegative and continuous function such that the function ω(s) = (ψ(|s|) − 1)|s|p is convex for the number p under consideration. Then the functional (4.6) is strictly uniformly convex for p ≥ 2 and quasiuniformly convex for 1 < p < 2 on the convex set D . Proof. Let
f (x, yk ) = q(x)ykp ψ(yk ),
yk ≥ 0,
and suppose that, for example, 1 < p < 2 . Then, for all uk , vk ∈ R , from inequality (3.8), as well as from the inequality q(x) ≥ q0 and the convexity of the function ω(s) = (ψ(|s|) − 1)|s|p , we obtain the estimate |uk |p + |vk |p + |uk |p (ψ(|uk |) − 1) + |vk |p (ψ(|vk |) − 1) f (x, |uk |) + f (x, |vk |) = q(x) 2 2 u k + vk p
+ q0 κ |uk |2−p + |vk |2−p |uk |p−1 − |vk |p−1 2 ≥ q(x) 2 u k + vk u k + vk p + q(x) ψ −1 2 2 p u k + vk
ψ uk + vk + q0 κ |uk |2−p + |vk |2−p |uk |p−1 − |vk |p−1 2 = q(x) 2 2 u k + vk
+ q0 κ |uk |2−p + |vk |2−p |uk |p−1 − |vk |p−1 2 . = f x, (4.7) 2 Hence it is clear that relation (4.2), with ϕ(s) = s , is satisfied. As was pointed out at the beginning of Sec. 4, this relation yields the condition of quasiuniform convexity (3.13) of the functional (4.6) on the convex set D , with ξ(t) = q0 κt . The proof of this theorem is similar to that in the case p ≥ 2 if, instead of (3.8), inequality (3.6) is used. Accordingly, the calculation of (4.7) is modified and, instead of (4.2), we obtain a version of relation (4.1). Corollary 4.4. If the functions q , ψ satisfy the assumptions of Theorem 4.3 and, moreover, ψ(s) is a nondecreasing function for s ≥ 0 , then the generating function f (x, y) = q(x)
m k=0
ykp ψ(yk )
, x ∈ T . Then, by Lemma 3.1, the of the functional (4.6) has properties (a)–(c) for y ∈ Rm+1 + functional (4.6) is lower semicontinuous and, by Theorem 3.2, (for p ≥ 2 ) or by Theorem 3.8 (for 1 < p < 2 ), in view of the result of Theorem 4.3, it has the H -property on D . As a specific example of such functionals (4.6), we can indicate m
Ω4 (z) = q(x)|Ak z(x)|p exp |Ak z(x)| dx k=0
T
and some others. In conclusion, it should be noted that, using a similar approach, one can also establish the H -property of the following analogs of the functionals (3.3), (4.3), and (4.6): Ω(z) = f (x, |z(x)|) dx, Ω(z) = f (x, ϕp (|z(x)|)) dx, T T q(x)|z(x)|p ψ(|z(x)|) dx Ω(z) = T
in the space Lp (T ) , p > 1 .
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ACKNOWLEDGMENTS This research was supported by the Russian Foundation for Basic Research under grant no. 0201-00044. REFERENCES 1. J. Diestel, Geometry of Banach Spaces—Selected Topics, no. 11, Lecture Notes in Math., vol. 485, Springer-Verlag, Berlin–Heidelberg, 1975. 2. L. V. Kantorovich and G. P. Akilov, Functional Analysis [in Russian], Nauka, Moscow, 1977. 3. A. S. Leonov, “The generalization of the maximal entropy method for solve ill-posed problems” Sibirsk. Mat. Zh. [Siberian Math. J.], 41 (2000), no. 4, 863–872. 4. F. P. Vasilev, Numerical Methods of Solution of Extremal Problems [in Russian], Nauka, Moscow, 1980. 5. M. M. Vainberg, Variational Methods of the Study of Nonlinear Operators [in Russian], Gostekhizdat, Moscow, 1956. 6. Functional Analysis [in Russian] (S. G. Krein, editor), SMB Series, Nauka, Moscow, 1972. 7. U. Amato and W. Hughes, “Maximum entropy regularization of Fredholm integral equations of the first kind,” Inverse Problems, 7 (1991), 793–808. 8. P. P. B. Eggermont, “Maximum entropy regularization for Fredholm integral equations of the first kind,” SIAM J. Math. Anal., 24 (1993), no. 6, 1557–1576. Moscow Institute of Engineering Physics (MIFI) E-mail :
[email protected]
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